j 

Z 

a 

^_ 

O 

(X 

hH 

J 

PQ 

O 

i  —  ( 

1     [L, 

^ 

i  o 

w 

>- 

u 

H 

I-H 

w 

01 

a: 

pq 

T 

UJ 

BIOLOGY 

G 


Hi U U L 


STATISTICAL    METHODS 


WITH  SPECIAL  REFERENCE  TO 


BIOLOGICAL    VARIATION. 


BY 


C.  B.  DAVENPORT,  PH.D., 

Instructor  in  Zoology  at  Harvard  University. 


FIRST   EDITION. 
FIRST   THOUSAND. 


NEW  YORK: 

JOHN   WILEY   &   SONS. 

LONDON:    CHAPMAN  &  HALL,   LIMITED. 

1899. 


Copyright,  1899, 

BY 

C.  B.  DAVENPORT. 

7 


ROBERT  DRUMMOND,   PRINTER,   NEW  YORK. 


PREFACE. 


THIS  book  has  been  issued  in  answer  to  a  repeated  call  for  a 
simple  presentation  of  the  newer  statistical  methods  in  their 
application  to  biology.  The  immediate  need  which  has  called 
it  forth  is  that  of  a  handbook  containing  the  working  formulae 
for  use  at  summer  laboratories  where  material  for  variation- 
study  abounds.  In  order  that  the  book  should  not  be  too 
bulky  the  text  has  been  condensed  as  much  as  is  consistent 
with  clearness. 

This  book  was  already  in  rough  draft  when  the  work  of 
Duncker  appeared  in  Roux's  Archiv.  I  have  made  much  use 
of  Duncker's  paper,  especially  in  Chapter  IV.  I  am  indebted 
to  Dr.  Frederick  H.  Safford,  Assistant  Professor  of  Mathe- 
matics at  the  University  of  Cincinnati  and  formerly  Instructor 
at  Harvard  University,  for  kindly  reading  the  proofs  and  for 
valuable  advice.  To  Messrs.  Keuff  el  and  Esser,  of  New  York, 
I  am  indebted  for  the  use  of  the  electrotypes  of  Figures  1  and  2. 
Finally,  I  cannot  fail  to  acknowledge  the  cordial  cooperation 
which  the  publishers  have  given  in  making  the  book  ser- 
viceable. 

C.  B.  DAVENPORT. 

BIOLOGICAL  LABORATORY  OF  THE  BROOKLYN  INSTITUTE, 

COLD  SPRING  HARBOR,  LONG  ISLAND, 

June  29,  1899. 

iii 


CONTENTS. 


CHAPTER  I. 
ON  METHODS  OF  MEASURING  ORGANISMS. 

PAGE 

Preliminary  Definitions 1 

Methods  of  Collecting  Individuals  for  Measurement 2 

Processes  Preliminary  to  Measuring  Characters 2 

The  Dermination  of  Integral  Variates— Methods  of  Counting 3 

The  Determination  of  Graduated  Variates— Method  of  Measurement..  4 

Straight  lines  on  a  plane  surface 4 

Distances  through  solid  bodies  or  cavities  — 4 

Area  of  plane  surfaces 4 

Area  of  a  curved  surface  5 

Form  of  a  plane  figure — 6 

Characters  occupying  three  dimensions  of  space 9 

Characters  having  weight 9 

Color  characters  9 

Marking-characters , 10 

CHAPTER  II. 

ON  THE  SERIATION  AND  PLOTTING  OF  DATA  AND  THE  FREQUENCY 
POLYGON. 

Seriation 11 

Plotting 12 

Method  of  loaded  ordinates 12 

Method  of  rectangles 13 

Certain  constants  of  the  Frequency  Polygon 13 

The  mean 13 

The  mode 14 

The  median  magnitude 14 

The  probable  error  of  the  mean 14 

The  index  of  variability 15 

The  probable  error  of  the  standard  deviation . .  15 

Average  deviation  and  probable  error 15 

Coefficient  of  variability 15 

V 


yi  COXTENTS. 

CHAPTER  III. 

THE  CLASSES  OF  FREQUENCY  POLYGONS. 

PAGE 

Classification 16 

To  classify  a  simple  frequency  polygon 16 

The  normal  curve  18 

To  compare  any  observed  curve  with  the  theoretical  normal 

curve 19 

The  index  of  abmodality 19 

To  determine  the  closeness  of  fit  of  a  theoretical  polygon  to  the 

observed  polygon 19 

The  normal  curve  as  a  binomial  curve 20 

Example  of  a  nearly  normal  curve 20 

Other  unimodal  frequency  polygons . 21 

Curves  of  limited  range 22 

Asymmetry  or  skewness 22 

To  compare  any  observed  frequency  polygon  of  Type  I  with  its 

corresponding  theoretical  curve 22 

To  compare  any  observed  frequency  polygon  of  Type  II  with  its 

corresponding  theoretical  curve 23 

To  compare  any  observed  frequency  polygon  of  Type  III  with  its 

corresponding  theoretical  curve 23 

To  compare  any  observed  frequency  curve  of  Type  IV  with  its 

corresponding  theoretical  curve 24 

Example  of  calculating  the  theoretical  curve  corresponding  with  ob- 
served data 25 

Multimodal  curves 26 

CHAPTER  IV. 
CORRELATED  VARIABILITY. 

General  principles 30 

Methods  of  determining  coefficient  of  correlation 32 

Galton's  graphic  method 32 

Pearson's  method 32 

Duncker's  brief  method 33 

Spurions  Correlation  in  Indices  35 

Heredity 35 

Uniparental  inheritance 36 

Biparental  inheritance 36 

Galton's  law  of  ancestral  heredity 37 

CHAPTER  V. 

SOME  APPLICATIONS  OF  STATISTICAL  BIOLOGICAL  STUDY. 

The  laws  of  variation....  , 38 

The  causes  of  variation 38 

Selection 38 

The  origin  of  species 38 


CONTENTS.  Vll 

PAGE 

The  definition  of  species 38 

Distinction  between  species  and  varieties  39 

Criterion  for  homology 39 

Prepotency 39 

SELECTED  BIBLIOGRAPHY  OF  WORKS  ON  THE  QUANTITATIVE  STUDY  OF 

ORGANISM 40 

EXPLANATION  OF  TABLES  43 

LIST  OF  TABLES. 

Table        I.  Formulas 53 

II.  Certain  constants  and  their  logarithms 54 

"         III.  Table  of  ordinates  of  normal  curve,  or  values  of  -=— 

2/o 

corresponding  to  values  of  — 55 

<r 

"         IV.  Table  of  values  of  the  normal  probability  integral  corre 
spending  to  values  of  —  ;  or  the  fraction  of  the  area  of 

<T 

the  curve  between  the  limits  0  and  -j or  0  and 56 

<r  <r 

**          V.  Table  of  Log  r  functions  of  p 57 

"         VI.  Table  of  reduction  of  linear  dimensions  from  common  to 

metric  system 59 

VII.  First  to  sixth  powers  of  integers  from  1  to  30 60 

"      VIII.  Squares,  cubes,  square-roots,  cube-roots  and  reciprocals,  60 

"         IX.  Logarithms  of  numbers 77 

"          X.  Logarithmic  sines,  cosines,  tangents  and  cotangents 104 


STATISTICAL  METHODS 

WITH   SPECIAL   REFERENCE  TO 

BIOLOGICAL 


CHAPTER  I. 
ON  METHODS  OF  MEASURING  ORGANISMS. 

Preliminary  Definitions. 

An  individual  is  a  segregated  mass  of  living  matter,  capable 
of  independent  existence.  Individuals  are  either  simple  or 
compound,  i.e.,  stocks  and  corms.  In  the  case  of  a  compound 
individual  the  morphological  unit  may  be  called  a  person. 

A  character  is  any  quality  common  to  a  number  of  in- 
dividuals. 

The  magnitude  of  a  character  is  a  quantitative  expression  of 
the  character. 

A  variate  is  a  single  magnitude-determination  of  a  charac- 
ter. 

A  class  includes  variates  of  the  same  or  nearly  the  same 
magnitude. 

Integral  variates  are  magnitude-determinations  of  charac- 
ters which  from  their  nature  are  expressed  in  integers.  Such 
magnitudes  are  determined  by  counting;  e.g. ,  the  number  of 
teeth  in  a  porpoise. 

Graduated  variates  are  magnitude-determinations  of  charac- 
ters which  do  not  exist  as  integers  and  which  may  conse- 


2  STATISTICAL   METHODS. 

quently   differ    in    different    individuals    by   any   degree  of 
magnitude  however  small;  e  g.t  the  stature  of  man. 

Methods  of  Collecting  Individuals  for  Meas- 
urement, 

In  collecting  a  lot  of  individuals  for  the  study  of  the  varia- 
bility of  any  character  undue  selection  must  be  avoided.  The 
rule  is: 

Having  settled  upon  tlie  general  conditions,  of  race,  sex, 
locality,  etc.,  which  the  individuals  to  be  measured  must  fulfil, 
take  the  individuals  methodically  at  random  and  without  possible 
selection  of  individuals  on  the  basis  of  the  magnitude  of  the 
character  to  be  measured.  If  the  individuals  are  simply  not 
consciously  selected  on  the  basis  of  magnitude  of  the  character 
they  will  often  be  taken  sufficiently  at  random. 

Processes  Preliminary  to  Measuring 
Characters. 

Some  characters  can  best  be  measured  directly;  e.g., "the 
stature  of  a  race  of  men.  Often  the  character  can  be  better 
studied  by  reproducing  it  on  paper.  The  two  principal 
methods  of  reproducing  are  by  photography  and  by  camera 
drawings. 

For  photographic  reproductions  the  organs  to  be  measured 
will  be  differently  treated  according  as  they  are  opaque  or 
transparent.  Opaque  organs  should  be  arranged  if  possible 
in  large  series  on  a  suitable  opaque  or  transparent  back- 
ground. The  prints  should  be  made  on  a  rough  paper  so 
that  they  can  be  written  on ;  blue-print  paper  is  excellent. 
This  method  is  applicable  to  hard  parts  which  may  be  studied 
dry;  e.g.,  mollusc  shells,  echinoderms,  various  large  arthro- 
pods, epidermal  markings  of  vertebrates  and  parts  of  the 
vertebrate  skeleton.  Shadow  photographs  may  be  made  of 
the  outlines  of  opaque  objects,  such  as  birds'  bills,  birds'  eggs, 
and  butterfly  wings,  by  using  parallel  rays  of  light  and  inter- 
posing the  object  between  the  source  of  light*  and  the  photo- 

*  A  Welsbach  burner  or  an  electric  light  are  especially  good.     Minute 


MEASUREMENT   OF    ORGANISMS.  3 

graphic  paper.  More  or  less  transparent  organs,  such  as 
leaves,  petals,  insect-wings,  and  appendages  of  the  smaller 
Crustacea,  may  be  reproduced  either  directly  on  blue-print 
paper  or  by  "solar  prints,"  either  of  natural  size  or  greatly 
enlarged.  For  solar  printing  the  objects  should  be  mounted 
in  series  on  glass  plates.  They  may  be  fixed  on  the  plate  by 
means  of  balsam  or  albumen  and  mounted  between  plates  either 
dry  or  in  Canada  balsam  or  other  permanent  mounting  media. 
Wings  of  flies,  orthoptera,  neuroptera,  etc.,  may  be  prepared 
for  study  in  this  way;  twenty-five  to  one  hundred  sets  of  wings 
being  photographed  on  one  sheet  of  paper,  say  16  X  20  inches 
in  size.  Microphotographs  will  sometimes  be  found  service- 
able in  studying  small  organisms  or  organs,  such  as  shells  of 
Protozoa  or  cytological  details. 

Camera  drawings  are  a  convenient  although  slow  method  of 
reproducing  on  paper  greatly  enlarged  outlines  of  microscopic 
characters,  such  as  the  form  and  markings  of  worms  and 
lower  Crustacea,  sponge  spicules,  bristles,  scales  and  scutes, 
plant-hairs,  cells  and  other  microscopic  objects.  In  making 
such  camera  drawings  a  low-power  objective,  such  as  ZeissA*, 
will  often  be  found  very  useful. 

The  Determination  of  Integral  Variates.— 
Methods  of  Counting. 

While  the  counting  of  small  numbers  offers  no  special  diffi- 
culty, the  counting  becomes  more  difficult  with  an  increase  of 
numbers.  To  count  large  numbers  the  general  rule  is  to  di- 
vide the  field  occupied  by  the  numerous  organs  into  many 
small  fields  each  containing  only  a  few  organs.  Counting 
under  the  microscope,  e.g.,  the  number  of  spines,  scales  or 
plant-hairs  per  square  millimetre,  may  be  aided  by  cross-hair 
rectangles  in  the  eyepiece.  The  number  of  blood-corpuscles 
in  a  drop  of  blood,  or  of  organisms  in  a  cubic  centimetre  of 
water,  have  long  been  counted  on  glass  slides  ruled  in  small 
squares. 

electric  lamps  such  as  are  fed  by  a  single  cell  give  sharp  shadows  of 
small  objects. 


STATISTICAL   METHODS. 


The    Determination  of  Graduated   Variates.— 
Methods  of  Measurement. 

Straight  lines  on  a  plane  surface  are  easily  meas- 
ured by  means  of  a  measuring-scale  of  some  sort.  The  meas- 
urement should  always  be  metric  because 
this  is  the  universal  scientific  system.  Vari- 
ous kinds  of  scales  may  be  obtained  of 
optical  companies  and  hardware  dealers, — 
such  as  steel  measuring  tapes,  graduated  to 
millimetres  (about  $1.00),  and  steel  rules 
(6  cm.  to  15  cm.)  graduated  to  ^  of  a  milli- 
metre. Steel  "spring-bow"  dividers  with 
milled-head  screw  are  useful  for  getting 
distances  which  may  be  laid  off  on  a  scale. 
Tortuous  lines,  e.g.,  the  contour  of  the 
serrated  margin  of  a  leaf  or  the  outer 
margin  of  the  wing  of  a  sphinx  moth,  may 
be  measured  by  a  map-measurer  ("Entfer- 
nungsmesser,"  Fig.  1),  supplied  at  artist's 
and  engineer's  supply  stores  at  about  $3.50. 
Distances  through  solid  bodies 
or  cavities  are  measured  by  calipers  of 
some  sort.  Calipers  for  measuring  diameters 
of  solid  bodies  are  made  in  various  styles. 
Micrometer  screw  calipers  ("  speeded") 
reading  to  one-huudredths  of  a  millimetre 
and  sold  by  dealers  in  physical  apparatus  for 
about  $5.00  are  excellent  for  determining  diameters  of  bones, 
birds'  eggs,  gastropod  shells,  etc.  Leg  calipers  for  rougher 
work  can  be  obtained  for  from  30  cents  to  $4.00.  The 
micrometer  "  caliper-square,"  available  for  inside  or  outside 
measurements  and  measuring  to  hundredths  of  a  millimetre, 
is  a  useful  instrument.* 

The  area  of  plane  surfaces,  as,  e.g  ,  of  a  wing  or  leaf, 
is  easily  determined  by  means  of  a  sheet  of  colloidin  scratched 
in  millimetre  squares.  By  rubbing  in  a  little  carmine  the 

*  Many  of  the  instruments  described  in  this  section  are  made  by  the 
Starrett  Co.,  Athol,  Mass.,  and  by  B  rown  and  Sharpe,  Providence,  tool 
cutters. 


FIG.  1. 


MEASUREMENT    OF    ORGANISMS.  5 

scratches  may  be  made  clearer.  The  number  of  squares 
covered  by  the  surface  is  counted  (fractional  squares  being 
mentally  summated)  and  the  required  area  is  at  once  obtained. 
If  the  area  has  been  traced  on  paper  it  may  be  measured  by 
the  planimeter  (Fig.  2).  This  instrument  may  be  obtained  at 


engineer's  supply  shops.  It  consists  of  two  steel  arms  hinged 
together  at  one  end;  the  other  end  of  one  arm  is  fixed  by  a 
pin  into  the  paper,  the  end  of  the  second  arm  is  provided  with 
a  tracer.  By  merely  tracing  the  periphery  of  the  figure  whose 
area  is  to  be  determined  the  area  may  be  read  off  from  a  drum 
which  moves  with  the  second  arm.  This  method  is  less 
wearisome  than  the  method  of  counting  squares. 

The  area  of  a  curved  surface,  like  that  of  the  elytra 
of  a  beetle  or  the  shell  of  a  clam,  is  not  always  easy  to  find. 
To  get  the  area  approximately,  project  the  curved  surface  on 
a  plane  by  making  a  camera  drawing  or  photograph  of  its 
outline.  By  means  of  parallel  lines  divide  the  outline  draw- 
ing into  strips  such  that  the  corresponding  parts  of  the  curved 
surface  are  only  slightly  curved  across  the  strips,  but  greatly 
curved  lengthwise  of  the  strips.  Measure  the  length  of  each 
plane  strip  and  divide  the  magnitude  by  the  magnification  of 
the  drawing.  Measure  also,  with  a  flexible  scale,  the  length 
of  the  corresponding  strip  on  the  curved  surface.  Then,  the 
area  of  any  strip  of  the  object  is  to  the  area  of  the  projection 
as  the  length  of  the  strip  on  the  object  is  to  the  length  of  its 
projection.  The  sum  of  the  areas  of  the  strips  will  give  the 
total  area  of  the  surface. 


6 


STATISTICAL    METHODS. 


The  form  of  a  plane  figure  of  irregular  outline  has 
been  expressed  qualitatively  by  botanists,  who  have  invented  a 
complicated  nomenclature  for  the  purpose;  this  is  reproduced 
in  part  here. 

Linear,  more  than  thrice  longer  than  wide  and  of  nearly  the 
same  breadth  throughout  (Fig.  3). 

Lanceolate,  more  than  thrice  longer  than  wide  and  tapering 
towards  one  or  both  ends  (Fig.  4). 

Oblong,  twice  to  thrice  as  long  as  broad  (Fig.  5). 

Elliptical,  of  the  shape  of  an  ellipse  with  an  eccentricity 
more  than  .5  (Fig.  6). 

Oval,  elliptical,  with  eccentricity  from  .5  to  .1. 

Orbicular,  nearly  circular,  with  eccentricity  less  than  .1. 

Ovate,  with  the  outline  of  a  hen's  egg,  one  end  broader  than 
the  other  (Fig.  7). 


A 


FIGS.  3-7. 

Cuneate  or  cuneiform,  wedge-shaped. 

Spatulate,  rounded  atone  end,  long  and  narrow  at  the  other, 
like  a  spatula. 

Acuminate,  tapering  to  an  angle  of  less  than  15°  (Fig.  8). 

Acute,  ending  in  an  angle  of  from  15°  to  90°  (Fig.  9). 

Obtuse.,  ending  in  an  angle  of  over  90°  (Fig.  10). 

Truncate,  terminating  as  though  cut  off  (Fig.  11). 

Retuse,  with  a  re-entering  obtuse  end  (Figs.  12-14). 

Serrate,  with  small  saw-like  teeth  (Fig.  15). 

Dentate,  with  larger,  more  obtuse  teeth  (Fig.  16). 

Crenate,  rounded  teeth  (Fig.  17). 

Repand,  wavy  margin,  teeth  broadly  rounded,  height  less 
than  breadth  (Fig.  18). 


MEASUREMENT  OF  ORGANISMS. 


U 


Sinuate,    still    stroDger  waves,    height   equals  or    exceeds 
breadth  (Fig.  19). 

Incised,  with  sharp,  deep  incisions  (Fig.  20). 


15 


17  18  19 

FIGS.  15-20. 


The  quantitative  expression  of  variation  in  these  forms  can 
usually  be  easily  obtained  by  using  an  index,  or  ratio  of  two 
dimensions. 

greatest  length 
Index  of  Lmearness,  --^--^-^^. 

greatest  length      . 
"      "  Lanceolateness,  j         -  T^J^T,'  also  angle  afo. 


greatest  length  area 

"  Oblongness,  greatest  breadtu.    ^     b^dth' 

(greatest  Igth.)  — (greatest  brdth.) 
^'  (greatest 

for  values  from  1  to  .50, 


Index  of  Ovalness, 


STATISTICAL   METHODS. 
(greatest  length)  —  (greatest  breadth) 


(greatest  length) 
for  values  from  .50  to  .1. 

(greatest  diam. ) — (greatest  brdth. ) 
Orbicularness,  —  — -, 

(greatest  diameter) 

for  values  from  .1  to  0. 

radius  of  curvature  of 
larger  end 

Ovateness  or  obovateness, -^ £ ri r  • 

radius  of  smaller  end 

diameter  at  £ 

Cuneateness,  T. -~,    or   angle  abc    (line  a-c 

diameter  at  f 

passing  through  middle  of  major  diameter). 

length  of  radius  of  curve  at  broad 

end  of  organ 

Spatulateness, — T —       — . 

transverse  diameter  of  narrow 

part  of  organ 

Acuminateness,  angle  abc  at  apex  (Fig.  8). 

Acuteness,  angle  abc  at  apex. 

Obtuseness,  angle  abc  at  apex  and  radius  of  curva- 
ture. 

Truncatedness,  angle  abc  at  apex  and  radius  of  curva- 
ture. 

cosine 

Retuseness, : —  of  |  angle  abc. 

2  X  sine 

Serrateuess,  number  of  teeth  per  linear  unit  of  edge, 

average  angle  of  tooth. 

Dentateness,  number  of  teeth,  average  angle  of  tooth, 
Crenateness,    number  of  waves,    average  radius  of 

curvature  of  waves. 

depth  of  waves 

Repandness,  = —  — ,  average  radius  of  cur- 

length  of  waves 

vature  of  waves. 

depth  of  waves 

Smuateness, : —  — ,  average  radius  of  cur- 

length  of  waves 

vature  of  waves. 

depth  of  incision 

1  Incisedness, —. j-. — — — . 

opening  of  incision 


MEASUREMENT    OF    ORGANISMS.  9 

Characters  occupying  three  dimensions  of 
space  may  be  quantitatively  expressed  by  volume.  The 
volume  of  water  or  sand  displaced  may  be  used  to  measure 
volume  in  the  case  of  solids.  The  volume  of  water  or  sand  con- 
tained will  measure  a  cavity.  Irregular  form  is  best  measured 
by  getting,  either  by  means  of  photography  or  drawings,  pro- 
jections of  the  object  on  one  or  more  of  the  three  rectangular 
fundamental  planes  of  the  organ,  and  then  measuring  these 
plane  figures  as  already  described.  Or  two  or  more  axes  may 
be  measured  and  their  ratio  found. 

Characters  having1  weight  are  easily  measured  ;  the 
only  precautions  being  those  observed  by  physicists  and 
chemists. 

Color    Characters.      Color  may  be  qualitatively  ex- 
pressed by  reference  to  named  standard  color  samples.     Such 
standard    color    samples    are    given    in    Ridgeway's    book, 
'  "  Nomenclature  of  Color,"  and  also  in  a  set  of  samples  manu- 
factured by  the  Milton  Bradley  Co.,  Springfield,  Mass. ,  costing 
6  cents.     The  best  way  of  designating  a  color  character  is  by 
means  of  the  color  wheel,  a  cheap  form  of  which  (costing  6 
cents)  is  made  by  the  Milton  Bradley  Co.     The  colors  of  this 
"top"  are  standard  and  are  of  known  wave-length  as  follows: 
Red,         656  to  G61  Green,  514  to  519 

Orange,  COG  to  611  Blue,     467  to  472 

Yellow,  577  to  582  Violet,  419  to  424. 

It  is  desirable  to  use  Milton  Bradley's  color  top  as  a  standard. 
Any  color  character  can  be  matched  by  using  the  elementary 
colors  and  white  and  black  in  certain  proportions.  The  pro- 
portions are  given  in  perceuts.  In  practice  the  fewest  possible 
colors  necessary  to  give  the  color  character  should  be  employed 
and  two  or  three  independent  determinations  of  each  should 
be  made  at  different  times  and  the  results  averaged.  So  far 
as  my  experience  goes  any  color  character  is  given  by  only 
one  least  combination  of  elementary  colors.  (See  Science, 
July  16,  1897.) 

When  there  is  a  complex  color  pattern  the  color  of  the 
different  patches  must  be  determined  separately.  In  case  of 
a  close  intermingling  of  colors,  the  colored  area  may  be  rapidly 
rotated  on  a  turntable  so  that  the  colors  blend  and  the  result- 


10  STATISTICAL    METHODS. 

ant  may  then  be  compared  with  the  color  wheel.  By  this 
means  also  the  total  melanism  or  albinism,  viridescence,  etc., 
may  be  measured. 

Marking-characters.  The  quantitative  expression  of 
markings  or  color  patterns  will  often  call  for  the  greatest 
ingenuity  of  the  naturalist.  Only  the  most  general  rules  can 
here  be  laid  down.  Study  the  markings  comparatively  in  a 
large  number  of  the  individuals,  reduce  the  pattern  to  its 
simplest  elements,  and  find  the  law  of  the  qualitative  variation 
of  these  elements.  The  variation  of  the  elements  can  usually 
be  treated  under  one  of  the  preceding  categories.  Find  in  how 
far  the  variation  of  the  color  pattern  is  due  to  the  variation  of 
some  number  or  other  magnitude,  and  express  the  variation  in 
terms  of  that  magnitude.  Remember  that  it  is  rarely  a  ques- 
tion whether  the  variation  of  the  character  can  be  expressed 
quantitatively  but  rather  what  is  the  best  method  of  express- 
ing it  quantitatively. 


SERIATIOK   AKD    PLOTTING   OF   DATA.  11 


CHAPTER  II. 

ON  THE  SERIATION  AND  PLOTTING  OF  DATA  AND  THE 
FREQUENCY  POLYGON. 

The  data  obtained  by  measuring  any  character  in  a  lot  of 
individuals  consists  either  of  amass  of  numbers  for  the  charac- 
ter in  each  individual  ;  or,  perhaps,  two  numbers  which  are  to 
be  united  to  form  a  ratio  ;  or,  finally,  a  series  of  numbers  such 
as  are  obtained  by  the  color  wheel,  of  the  order  :  W  40%,  tf 
(Black)  38$,  T  12%,  O  Wo.  The  first  operation  is  the  simplifi- 
cation of  data.  Each  variate  must  be  represented  by  one 
number  only.  Consequently,  quotients  of  ratios  must  be  de- 
termined and  that  single  color  of  a  series  of  colors  which  shows 
most  variability  in  the  species  must  be  selected,  e.g.,N. 

The  process  of  seriation,  which  comes  next,  consists  of  the 
grouping  of  similar  magnitudes  into  the  same  magnitude 
class.  The  classes  being  arranged  in  order  of  magnitude, 
the  number  of  variates  occurring  in  each  class  is  determined. 
The  number  of  variates  in  the  class  determines  the  frequency 
of  the  class. 

The  method  of  seriation  may  be  illustrated  by  two  examples  ;  one  of 
integral  variates,  and  the  other  of  graduated  variates. 

Example  1.  The  magnitude  of  21  integral  variates  are  found  to  be  as 
follows  :    12,  14,  11,  13,  12,  12,  14,  13,  12,  11,  12,  12,  11,  12,  10,  11,  12,  13,  12, 
13,  12,  12t    In  seriation  they  are  arranged  as  follows  : 
Classes:        10,11,12,13,14. 
Frequency :    1,    4,  11,    4,    2. 

Example  2.  In  the  more  frequent  case  of  graduated  variates  our  mag- 
nitudes might  be  more  as  follows  : 

3.2  4.5  5.2  5.6  6.0 
3.8           4.7           5.2           5.7  6.2 
4.1           4.9           5.3           5.8  6.4 

4.3  5.0  5.3  5.8  6.7 
4.3           5.1           5.4           5.9           7.3 

In  this  case  it  is  clear  that  our  magnitudes  are  not  exact,  but  are  merely 
approximations  of  the  real  (forever  unknowable)  value.  The  question 


]2  STATISTICAL    METHODS. 

arises  concerning  the  inclusiveiiess  of  a  class — the  class  range.  An 
approximate  rule  is  :  Make  the  classes  only  just  large  enough  to  have 
no  or  very  few  vacant  classes  in  the  series.  Following  this  rule  we  get 


Classes  ... 
Frequency 


3.0-3.4; 

3.5-3.9; 

4.0-4.4; 

4.5-4.9; 

5.0-5.4; 

3.2 

3.7 

4.2 

4.7 

5.2 

*] 

o 

3 

4 

5 

1 

1 

3 

3 

7 

5.5-5.9; 

6.0-6.4; 

6.5-6.9; 

V.0-7.4; 

5.7 

6.2 

6.7 

7.2 

6 

7 

8 

9 

5 

3 

1 

1 

Frequency 

The  classes  are  named  from  their  middle  value,  or  better,  for  ease  of 
subsequent  calculations,  by  a  series  of  small  integers  (1  to  9). 

In  case  the  data  show  a  tendency  of  the  observer  towards  estimating 
to  the  nearest  round  number,  like  5  or  10,  each  class  should  include  one 
and  only  one  of  these  round  numbers. 

As  Fechner  ('97)  has  pointed  out,  the  frequency  of  the  classes  and  all 
the  data  to  be  calculated  from  the  series  will  vary  according  to  the 
point  at  which  we  begin  our  seriation.  Thus  if,  instead  of  beginning  the 
series  with  3.0  as  in  our  example,  we  begin  with  3.1  we  get  the  series  : 


Classes  •) 

3.1-3.5; 

3.6-4.0; 

4.1-4.5; 

4.6-5.0; 

5.1-5.1 

3.3 

3.8 

4.3 

4.8 

3.5 

Frequency 

1 

1 

4 

3 

6 

Classes  .  .  .  .  .) 

5.6-6.0; 

5.8 

6.1-6.5; 
6.3 

6.6-7,0; 
6.8 

7.1-7.5; 
7.3 

Frequency 

6 

2 

1 

1 

which  is  quite  a  different  series.  Fechner  suggests  the  rule:  Choose  such 
a  position  of  the  classes  as  will  give  a  most  normal  distribution  of  fre- 
quencies. According  to  this  rule  the  first  distribution  proposed  above 
is  to  be  preferred  to  the  second. 

In  order  to  give  a  more  vivid  picture  of  the  frequency  of 
the  classes  it  is  important  to  plot  the  frequency  polygon. 
This  is  done  on  coordinate  paper.* 

A  different  method  should  be  adopted  according  as  integral 
or  graduated  variates  are.unler  consideration.  In  the  case  of 
integral  varia'cs  proceed  as  follows  :  At  equal  intervals  along 
a  horizontal  line  (axis  of  X)  draw  a  series  of  (vertical)  ordinates 
whose  successive  heights  shall  be  proportional  to  the  frequency 
of  the  classes.  Join  the  tops  of  the  ordinates.  Thus  for  the 
example  given,  the  curve  will  be  as  shown  in  Fig.  21.  This 
method  of  drawing  the  frequency  polygon  is  known  as  the 
method  of  loaded  ordinates. 

*  This  paper  may  be  obtained  at  any  artists'  supply  store. 


SERIATION    AND    PLOTTING    OF    DATA. 


13 


In  the  case  of  graduated  variates  proceed  as  follows :  Lay 
off  along  a  horizontal  line  equal  contiguous  spaces  each  of 
which  shall  represent  one  class,  number  the  spuces  in  order 


9 


11 


13 


15 


12 
FIG.  21. 

from  left  to  right  with  the  class  magnitudes  in  succession, 
and  erect  upon  these  bases  rectangles  proportionate  in  height 
to  the  frequency  of  the  respective  clashes  (Fig.  22). 


~~i    ,    , 

3.0         3.5         4,0        4,5         5.0         5.5         e,0         6.5         7.0         7.5 
FIG.  2-2. 

This  method  of  drawing  the  frequency  polygon  is  known  as 
the  method  of  rectangles.  If  the  tops  of  the  middle 
ordinates  of  successive  contiguous  rectangles  be  connected  by 
an  oblique  line  a  polygon  made  up  of  trapezia  is  obtained. 
The  outline  of  the  polygon  will  be  fairly  close  to  that  of  a 
curve  passing  through  the  tops  of  the  central  ordinates  of  the 
rectangles. 

CERTAIN  CONSTANTS  OF  THE  FREQUENCY  POLYGON. 

After  the  data  have  been  gathered  and  arranged  it  is  neces- 
sary to  determine  the  law  of  distribution  of  the  variates.  To 
get  at  this  law  we  must  first  determine  certain  constants. 

The  mean  (M )  is  the  abscissa  of  the  centre  of  gravity  of 
the  variates  or  of  the  frequency  polygon.  It  is  found  by 
the  formula 


in  which  V  is  the  magnitude  of  any  class  ;  /  its  frequency  ; 


14  STATISTICAL   METHODS. 

2  indicates  that  the  sum  of  the  products  for  all  classes  into 
frequency  is  to  be  got,  and  n  is  the  number  of  variates, 

Thus  in  the  last  example  : 
M   =  (3.2  X  1  +  3.7  X  1  +  4.2  X  3  +  4.7  X  3  +  5.2X  7  +  5.7  X  5  +  6.2  X  3 

+  6.7  X  1  +  7.2  X  1)  -4-25  =  5.24, 
or 

Ml  =  (lxl+2Xl+3x3+4x3-f5x7+6x5+7x3-f-8Xl-h9xl)-*-25  =  5.08, 
M    =  5.2*  +  .08  (5.7  -  5.2)  =  5.24 

A  still  shorter  method  of  finding  If  is  given  on  page  17. 

The  mode  is  the  class  with  the  greatest  frequency. 
In  the  example,  the  mode  is  5.2. 

The  median  magnitude  is  one  above  which  and  below 
which  50$  of  the  variates  occur.  It  is  such  a  point  on  the  axis 
of  X  of  the  frequency  polygon  that  an  ordinate  drawn  from  it 
bisects  the  polygon  of  rectangles  or  the  continuous  curve,  but 
not  the  polygon  of  loaded  ordinates. 

To  find  its  position:  Divide  the  variates  into  three  lots:  those  less  t han 
the  middle  cZass,  of  which  the  total  number  is  a;  those  of  the  middle 
class,  b;  and  those  greater,  c.  Then  a  +  b  -f  c  =  n  =  the  total  number 
of  variates.  Let  I'  =  the  lower  limiting  value  of  the  middle  class,  and 
I"  =  the  upper  limiting  value,  and  let  x  =  the  abscissal  distance  of  the 
median  ordinate  above  the  lower  limit  or  below  the  upper  limit  of  the 
median  class  according  as  x  is  positive  or  negative.  Then  £?i  -  a  :  b  = 
x\l"  —  V  ichen  x  is  positive,  or  \n  —  c  :  b  =  x  :  I"  —  I'  when  x  is  negative. 

Thus  in  the  last  example :  12.5  —  8  :  7  =  x  :  0.5;  x  =  .32;  the  median 
magnitude  =  5.0  +  .32  =  5.32.  Or  12.5  -  10  :  7  =  -  x  :  0.5;  x  =  -  .18;  the 
median  magnitude  =  5.5  —  .18  =  5.32.  (Cf.  p.  11.) 

Every  determination  of  a  constant  ot  the  frequency  polygon 
is  an  approximation  only  to  the  true  value  of  the  constant. 
The  closeness  of  the  approximation  to  the  truth  is  measured  by 
the  so-called  probable  error  of  the  determination.  This  is  a 
pair  of  values  lying  one  above  and  one  below  the  value  deter- 
mined. We  can  say  that  tfrere  is  an  even  chance  that  the  true 
value  lies  between  these  limits  ;  the  chances  are  4  to  1  that  the 
true  value  lies  within  twice  these  limits,  and  19  to  1  that  it  lies 
within  thrice  these  limits. 

The  probable  error  of  the  mean  is  given  by  the  for- 
mula 

±  0.6745  X  Sta"dard  deviatioa  [see  below]    =  ±  0-674g_^ 
|/number  of  variates  \/n 

It  will  be  seen  that  the  probable  error  is  less,  that  is,  that 
the  result  is  more  accurate,  the  greater  the  number  of  variates 
*  5.2  is  the  true  class  magnitude  corresponding  to  the  integer  5. 


SERIATIOtf    AKD   PLOTTING    OF    DATA.  15 

measured,  but  the  accuracy  does  not  increase  in  the  same  ratio 
as  the  number  of  individuals  measured,  but  as  the  square  root 
of  the  number.  The  probable  error  of  the  mean  decreases  as 
the  standard  deviation  decreases. 

The  index  of  the  variability,  cr,  of  the  variates  when 
they  group  themselves  about  one  mode  is  found  by  adding 
the  products  of  the  squared  deviation-from-the-mean  of  each 
class  multiplied  by  its  frequency,  dividing  by  the  total 
number  of  variates,  and  extracting  the  square  root  of  the 
quotient,  thus  : 


sum  of  [(deviation  of  class  from  mean)8 

X  frequency  of  class] 

number  of  variates 


n 

This  measure  is  known  as  the  standard  deviation. 
The  probable  error  of  the  standard  deviation  is 

standard  deviation  cr 

±  0.674o — =  ±  0.6745  — . 

y  2  X  number  of  variatts  \/2n 

Other  Indices  of  Variation  are  the  average  deviation,  or  aver- 
age departure, which  is  found  thus: 

of  [deviations  of  class  from  mean  x  frequency] 

number  of  variates 
The  probable  error  is  the  distance  from  the  mode  of  that  ordinate 
which  exactly  bisects  the  half  curve  OMX  or  OMX*,  Fig.  23;  it  is  equal  to 
0.6745  X  standard  deviation  =  0.6745<r.     Neither  of  these  last  two  indices 
of  variation  is  as  good  as  the  standard  deviation  when  n  is  rather  small. 

The  standard  deviation,  like  the  other  indices  of  variation, 
is  a  concrete  number,  being  expressed  in  the  same  units  as  the 
magnitudes  of  the  classes.  The  standard  deviation  of  one  lot 
of  variates  is  consequently  not  comparable  with  the  S.  D.  of 
variates  measured  in  other  units.  It  has  been  proposed  to  re- 
duce the  index  of  variation  to  a  concrete  number,  independent 
of  any  particular  unit,  by  dividing  the  index  of  variation  of  any 
variates  by  the  mean  ;  the  quotient  multiplied  by  100  is  called 

the  coefficient  of  variability.    In  a  formula,  GV  —  ^. 
(Pearson,  '96  ;  Brewster,  '97. ) 


16  STATISTICAL   METHODS. 


CHAPTER   III. 

THE  CLASSES  OF  FREQUENCY  POLYGONS. 

The  plotted  curve  may  fall  into  one  of  the  folio  wing  classes  : 

A.  Unimodal. 

I.  Simple. 

1.  Range  unlimited  in  both  directions: 

a.  Symmetrical.     The  normal  curve. 
It.  Unsymmetrical  (Pearson's  Type  IV). 

2.  Range  limited   in   one    direction,    together    with 

skewness  (Type  III). 

3.  Range  limited  in  both  directions  : 

a.  Symmetrical,  Type  II. 

b.  Unsymmetrical,  Type  I. 
II.  Complex. 

B.  Multi modal. 

The  classification  of  any  given  curve  is  not  always  an  easy 
task.  Whether  the  curve  is  unimodal  or  multimodal  can  be 
told  by  inspection.  "Whether  any  unimodal  curve  is  simple 
or  complex  cannot  be  told  by  any  existing  methods  without 
great  labor  and  uncertainty  in  the  result. 

Complex  curves  may  be  classified  as  follows  : 

1.  Composed  of  two  curves,  whose  modes  are  different  but  so  near  that 
the  component  curves  blend  into  one  ;   such  curves  are  usually  unsym- 
metrical. 

2.  The  sum  of  two  curves  having  the  same  mode  but  differing  varia- 
bility. 

3.  The  difference  of  two  curves  having  the  same  mode  but  differing 
variability. 

If  the  material  is  believed  to  be  homogeneous  and  the  curve 
is  unimodal  it  is  probably  simple  and  its  classification  may  be 
carried  further. 

For  classification  the  rule  is  as  follows  :  Determine  the  mean 
of  the  magnitudes.  Take  a  class  near  the  mean  (call  it  Vm) 


THE  CLASSES  OF  FKEQUENCY  POLYGONS.    17 

as  a  zero  point ;  then  the  departure  of  all  the  other  classes 
will  be  -  1,  —  2,  -  3,  etc.,  and  +  1,  -f  2,  -f  3,  etc. 

Add  the  products  of  all  these  departures  multiplied  by  the 
frequency  of  tbe  corresponding  class  and  divide  by  n\  call 
the  quotient  v\. 

Add  the  products  of  the  squares  of  all  the  departures  multi- 
plied by  the  frequency  of  the  corresponding  class  and  divide 
by  n\  call  the  quotient  v*. 

Add  the  products  of  the  cubes  of  all  the  departures  multiplied 
by  the  frequency  of  the  corresponding  class  and  divide  by  n\ 
call  the  quotient  v*. 

Add  the  products  of  the  fourth  powers  of  all  the  departures 
multiplied  by  the  frequency  of  the  corresponding  class  and 
divide  by  n\  call  the  quotient  r4.  Or, 

Vm) 
-  =  departure  of   Vm  from  mean.      Vm  being 

known,  M  may  be  found  [M  =  Vm  +  v\\\  * 

_  2(  v-  vmy 


The  values  rlf  ?'a,  v3f  r4,  are  called  respectively  the  first, 
second,  third,  and  fourth  moments  of  the  curve  about  Vm. 

To  get  the  moments  of  the  curve  about  the  mean,  either  of 
two  methods  (A  or  B)  will  be  employed.  Method  A  is  used 
when  integral  variates  are  under  consideration  ;  method  B 
when  we  deal  with  graduated  variates. 

(A)  To  find  moments  in  case  of  integral  variates: 
Hi  =  0; 

Mi  =  r*  —  v^\ 

ju3  —  r3  —  Srirt  -f  2ris; 

J*4    =    f4    —    4^i^3    +    6^iVa    —    3^!4. 

(B)  To  find  moments  in  case  of  graduated  variates  : 

*  This  is  the  short  method  of  finding  M  referred  to  on  page  14. 


18 


STATISTICAL   METHODS. 


-J- 


+ 


Also- 


F  =  6  +  3/5;  -  2y52  =  the  "  critical  function." 
Now  the  classification  of  any  empirical  curve  depends  upon 
the  value  of  its  critical  function,  F. 

j  (  fii  >  0,  curve  is  of  Type  I. 

When  F  is  positive  and  J  ',         '  V    -  ^        TT 

(  /Ji  =  0,  ^2  <  3,  curve  is  of  Type  II. 

"      F  =  0  and  j  ^  >  °'  ^a  >  3>  °UrVe  iS  °f  Type111* 

j  /3{  =  0,  /?2  —  3,  curve  is  normal. 

"     ^  is  negative  and    ft,  >  0,  /?a  >  3,  curve  is  of  Type  IV. 
An  important  relation  to  be  referred  to  later  is 


in  which  s  is  an  unknown,  positive  number. 

M 


/ 

\ 

I 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

[ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

5      4      3      2 

I 

L      0      1      2      3      4      5 
^IG.  23. 

THE  NORMAL  CURVE. 

The  normal  curve  is  symmetrical  about  the  mode  ;  con- 
sequently the  mode  and  the  median  and  mean  class  coincide. 
The  mathematical  formula  of  the  normal  curve,  a  formula 
which  one  does  not  have  to  understand  in  order  to  make  use 
of  it,  is 

a.  1 


THE  CLASSES  OF  FREQUENCY  POLYGONS.    19 

This  formula  gives  the  value  of  any  ordinate  y  (or  any 
class)  at  any  distance  x  (measured  aioug  the  base,  X,  X',  of 
Fig,  23)  from  the  mode,  e  is  a  constant  number,  2.71828,  the 
base  of  the  Naperian  system  of  logarithms,  a  is  the  total  area 
of  the  curve  or  number  of  variutes,  and  or  is  the  Standard 
Deviation,  which  is  constant  for  an}'  curve  and  measures  the 
variability  of  the  curve,  or  the  steepness  of  its  slope. 

To  compare  any  observed  curve  with  the  theo- 
retical normal  curve  we  can  make  use  of  tables.  For 
the  case  of  a  polygon  of  integral  variates  the  theoretical  fre- 
quency of  any  class  at  a  deviation  —  from  the  mean  can  be 

taken  directly  from  Table  III.  Here  x  is  the  actual  deviation 
from  the  mean  expressed  in  the  unit  of  the  maximum,  and  cr 
is  the  standard  deviation. 

For  the  case  of  a  polygon  of  graduated  variates  built  up  of 
rectangles  representing  the  relative  frequency  of  the  variates, 
Table  IV  gives  the  relation  of  the  actual  to  the  theoretical 

y> 

number  of  individuals  occurring  between  the  values  -j and 

— .  By  looking  up  the  given  values  of  —  the  correspond- 
ing theoretical  percentage  of  variates  between  the  limits 
-| and '—  will  be  found  directly.  The  ratio  —  maybe 

called  the  Index  of  Abmodality. 

The  normal  curve  may  preferably  be  employed  even  when 
/?!  is  not  exactly  equal  to  0,  nor  /?3  exactly  equal  to  3,  nor  F 
exactly  equal  to  0.  Use  the  normal  curve  when 

Rv   *   __   0T,  4 

F  X  /<33  <  ±  1  and  -  l—^  -  =  1  ±  .2 

To  determine  the  closeness  of  fit  of  a  theoreti- 
cal polygon  to  the  observed  polygon.  There  are 
two  methods  according  as  the  variates  are  (A)  integral  or  (B) 
graduated. 

(A)  Find  for  each  class  the  percentage  which  the  difference 
between  the  theoretical  value  y  and  the  observed  frequency 
/is  of  the  frequency,  and  fiud  the  average  of  these  percent- 
ages, which  is  the  index  of  closeness  of  fit  sought. 


20  STATISTICAL   METHODS. 

(B)  Subtract  in  order  each  theoretical  value  of  y  from  the 
corresponding  observed  value,  regarding  signs.  Call  the  dif- 
ference Si.  Whenever  in  the  successive  values  of  61  there  is 
a  change  of  sign,  divide  the  product  of  these  successive  values 
of  di,  in  pairs,  by  their  sum.  Call  this  value  &2;  make  its 
sign  always  minus.  Then  the  difference  between  the  two 
polygons  in  per  cent  of  one  of  them  is  given  by  the  equation 

^1  +  (_a,) 

3» 

where  £1  is  summated  without  regard  to  sign,  and  n  equals 
the  total  number  of  variates.  This  is  the  method  of  Duncker, 
'98.  It  may  be  considered  a  sufficient  agreement  between 

observation  and  calculation  when  A  <  — -%. 

\'n 

THE  NORMAL  CURVE  OF  FREQUENCY  AS  A  BINOMIAL 
CURVE. 

The  normal  curve  may  also  be  expressed  by  the  binomial 
formula  (p  +  <?)*,  where  p  =  \,q  =  -|,  and  I  is  the  number 
of  terms,  less  1,  in  the  expansion  of  the  binomial  ;  hence 
approximately  the  number  of  classes  into  which  the  magni- 
tudes of  the  variates  should  fall.  If  the  standard  deviation  be 
known,  I  may  be  found  by  the  equation 

I  =  4  X  (Standard  Deviation)2  =  4cr*. 


Example  of  (nearly)  normal  curve.    Number  of  spines  in 
dorsal  fin  of  Acerina  cernua,  L.  (Duncker,  '99,  p.  177). 


F 

F-Fm 

/    /(F-Fm)  /(F-Fm)2  /(F-Fm)3  /(F-l 

11 

-  3 

1 

-   3 

9 

-  27 

81 

12 

2 

2 

-   4 

8 

-  16 

32 

13 

-  1 

189 

-  189 

189 

-  189 

189 

14 

0 

1234 

0 

0 

0 

0 

15 

1 

454 

454 

454 

454 

454 

16 

2 

20 

40 

80 

160 

320 

1900  298  740  382  1076 


M=  Vm  +  vl  =  14  +  0.1568  =  14.1568. 
/*,  =  0.3895  -  0.1568*  =    0.3650. 

/m3  =  0.2011  -  3  X  0.1568  X  0.3895  -f  2  X  0.15683  =  0.0257. 


THE    CLASSES    OF   FREQUENCY    POLYGONS.          21 

f*4  =  0.5663  -  4X0.1568  X  0/2011  -f  6  X  0.15682  x  0.3895  -  3  X  0.1568*  =  0.4929. 


p  =  6  +  .04074  -  7.3996  =  -  1.3589.        F  .  ^  =  1.3589  X  0.3653  =  <066. 


v4  0.5663 

n  1900 

Maximum  frequency  =  —  =  -=  =  1255. 

<r  |/2ir       -6041  X  V2ir 

Although  somewhat  more  closely  of  Type  IV  (see  page  18)  than  of 
the  Normal  Type,  this  example  may  be  treated  as  Normal. 

The  difference  between  it  and  the  normal  is  found  below  to  be  1 .39^. 

To  illustrate  the  method,  and  in  accordance  with  Duncker's  example, 
A  is  here,  exceptionally,  calculated  by  rule  page  20. 


V—  M 

f 

y 

&i 

8a 

a 

-  3.157 

5.23 

1 

0.0 

+  1 

-  2.157 

3.57 

2 

2.1 

-  0.1 

-  0.09 

»  1.157 

1.92 

189 

200.4 

-  10.6 

-0.157 

.26 

1234 

1213,0 

+  21.0 

-  7.04 

+  0.843 

1.40 

454 

474.0 

-20.0 

-  10.24 

+  1.843 

3.50 

20 

11.9 

+  8.1 

-  5.76 

1901.5  60.8  23.1 


The  values  of  y  in  the  table  above  are  calculated  from  the  formula 
y  =  yQ  .  e-^2/20"2.  The  sum  of  the  theoretical  y  values  should  equal  the 
total  number  of  variates. 

OTHER  UNIMODAL  FREQUENCY  POLYGONS. 

The  formulas  of  the  remaining  four  types  of  unimodal  simple  fre- 
quency polygons  have  a  family  resemblance  with  the  formula 

_    *2 
y  =  y  e      2<7"a 

of  the  normal  curve.    They  are  as  follows: 
Curve  of  limited  range  on  both  sides: 

Unsymmetrical,    y  =  y0(l  +  ^p/1  (l  -  ^-j**        T^P6  L 

Symmetrical,          y  =  y0(l  -  '^)"\  Type  II. 

Curve  of  range  limited  on  one  side: 

Unsymmetrical,     y  =  yQ\l  +~)    e  ,  Type  III. 


22  STATISTICAL    METHODS. 


Curves  of  unlimited  range  on  both  sides: 

Unsymmetrical,    y—yQ  cos  0~    e        ,  where  tan  0  =  -,    Type  IV. 


[Symmetrical,      2/  =  2/0e  ,  the  normal  curve.] 

In  these  formulas  : 

2/0  =  modal  ordinate,  to  be  especially  reckoned  for  each  type. 

y   =  the  length  of  the  ordinate  (or  area  of  rectangle)  located  at 

the  distance  x  from  yQ. 
a  =  a  part  of  the  abscissa-axis  XXT  expressed  in  units  of  the 

•  classes. 
e   =  the  base  of  the  Naperian  system  of  logarithms,  2.71828. 

Curves  of  limited  range  are  theoretically  different  from  the 
normal  curve,  which  theoretically  applies  to  cases  where  the  classes 
have  an  infinite  range  above  and  below  the  mean.  Such  an  infinite 
range  is  rare  in  biological  statistics,  although,  as  stated,  the  normal 
curve  often  fits  observational  curves  very  closely.  The  range  in 
biological  statistics  may  be  limited  at  both  extremes.  Thus,  the  ratio 
of  carapace  length  to  total  length  of  the  lobster  is  limited  between  0 
andl. 

The    range    may    be   limited  on    one    side    only.        Thus    the    ratio 

Antero-Post.  Diam.     . 

-rr  -  -  -  —  -  of  a  bivalve  shell  may  conceivably  range  from  0  to 
Dorso-Veut.  Diam. 

oo.    The  forms  of  the  molluscan  genera  Pinna  (or  Malleus)  and  Solen 
approach  such  extremes. 

Asymmetry  or  skewness  is  found  in  Type  I  (of  which  Type  II  is 
the  symmetrical  limit),  Type  III  and  Type  IV.  In  skew  curves  the  mode 
and  the  mean  are  separated  from  each  other  by  a  certain  distance,  d. 
Asymmetry  is  measured  by  a  factor 


the  result  has  the  same  sign  as  /xs. 
In  Type  I,  A  =  Y*  Vftj^f. 
"  ll  III,  A=  14  VW- 


To  compare  any  observed  frequency  polygon  of  Type 
I  with  its  corresponding  theoretical  curve. 


THE   CLASSES   OF   FREQUENCY    POLYGONS.          23 

To  find  «x,  aa,  m^  wa,  y0. 

The  total  range,  6,  of  the  curve  (along  the  abscissa  axis)  is  found  by 
the  equation 


aj  and  aa  are  the  ranges  to  the  one  side  and  the  other  of  yQ] 
a^  =  ^b  —  ds);  d  =  irA  =  VVa  .  A\ 


aa  =  6  — 


yQ   =  _    .   I1L1 L_ZI2« m  -vi      '       "      '     "'     . 

To  solve  this  equation  it  will  be  necessary  to  determine  the  value  of 
each  parenthetical  quantity  following  the  r  sign  and  find  the  corre- 
sponding value  of  r  from  Table  V.  It  is,  however,  sometimes  easier  to 
calculate  the  value  of  yQ  from  the  following  approximate  formula: 

(m, m*^ gh ^ae      »»i       ma       n^       wia   ^ 

O  I/27TWI     »J 

With  these  data  the  theoretical  curve  of  Type  I  may  be  drawn.  Fre- 
quency polygons  of  Type  I  are  found  in  biological  measurements. 

To  compare  any  observed  frequency  polygon  of  Type 
II  with  its  corresponding  theoretical  curve. 


»-*(»- D" 


This  equation  is  only  a  special  form  of  the  equation  of  Type  I  in  which 
a!  =  «a  and  ??*!  =  ma. 

As  from  page  17,  /3j  =  0  in  Type  II,  b  =  2o-  V«  -f  1  ;  since  the  curve  is 
symmetrical,  d  =  0,  and 

b  &   T(m  +  1.5) 

—  -  —    -  - 


The  r  values  will  be  found  from  Table  V. 

An  approximate  formula  for  y0  is  given  by  Duncker  as  follows: 


s  -  1 


1/27T    V(s  +  l)(s  -  2) 

To  compare  any  observed  frequency  curve  of  Type  III 
with  Its  corresponding  theoretical  curve. 


24  STATISTICAL    METHODS. 

The  range  at  one  side  of  the  mode  is  infinite;  at  the  other  is  found 
by  the  formula 

a  =  o-4  ~  ^i.  =  a1  ~/2    (for  Type  III). 
2  VjSj 

a         a  Of.          »J>  +  ] 

Also,  p  =  —  ==—-;       2/o  =  — 


The  value  of  P  corresponding  to  p  +  1  can  be  got  from  Table  V, 
Appendix. 

To  compare  any  observed  frequency  curve  of  Type  IV 
with  its  corresponding  theoretical  curve. 

This  is  the  commonest  type  of  biological  skew  curves. 

y  =  y0(cos9}2m.e-vd. 

9  is  &  variable,  dependent  upon  x  as  shown  in  the  equation 
x  =  a  tan  9. 

The  factor  (cos  0)2m  following  yQ  indicates  that  the  curve  is  not  calcu 
lated  from  the  mean  ordinate  (If),  or  the  mode  (M  —  d),  but  that  the 
zero  ordinate  is  at  M  —  md;  or  at  a  distance  m  X  d  from  the  mean. 


4 

/t/ZiIs(s 2)  i/07 

v  — ^-i— 1,    with  the  opposite  sign  to  pB; 


0(arc  of  circle)  =  ^^; 

^__L_ 


a    /T 

2/o  =  -  V  5Z  • 


—  angle  whose  tangent  is  — . 


*  The  foregoing  value  is  approximate  and  is  applicable  when,  as  is 
usually  the  case,  s  is  greater  than  2.  The  exact  value  is  given  by  Pear- 
son as 

a  <,&•* 

Vo  =  ~ 


a         /»7T 

/      (sin 
t/0 


9fevddd 


the  formula  for  reducing  which  is  to  be  gained  from  the  integral  calcu- 
lus. 


THE    CLASSES   OF   FREQUENCY    POLYGONS.          25 


Example  of  calculating  the  theoretical  curve  corre- 
sponding with  observed  data.  (Fig.  24.) 

Distribution  of  frequency  of  glands  in  the  right  fore  leg  of  2000  female 
swine  (integral  variates): 

Number  of  glands      0        1        2l      3     •  4       5        6        7        8        9        10      V 
Frequency 15      20p    365    482    U14    277\  134     72      22      8         2 

Assume  the  axis  yy'  (  Vm)  to  pass  through  ordinate  4,  then: 
V         V  -  Vm        f      f(V—Vni)    f(V—Vm)*    f(V—V™)*    f(V—} 


0 

—  4  - 

15 

—  60 

240 

—  960 

3840 

1 

—  3 

209 

—  627  ' 

1881 

—  5643 

16929 

2 

2 

365 

—  730 

1460 

—  2920 

5840 

3 

—  1 

482 

—  482 

482 

—  482 

482 

4 

0 

414 

0 

0 

0 

0 

5 

1 

277 

277 

277 

277 

277 

6 

2 

134 

268 

536 

1072 

2144 

7 

3 

72 

216 

648 

1944 

5832 

8 

4 

22 

88 

352 

1408 

5632 

9 

5 

8 

40 

200 

1000 

5000 

10 

6 

2 

12 

72 

432 

2592 

2        2000  —998  6148  —3872  48568 

vl=  —    998  -T-  2000  =  —    .499. 

va  =       6148  -*-  2000  =       3.074. 

v,  =  —  3872  -4-  2000  =  —  1.936. 

»>4  =     48568  -f-  2000  =     24.284. 

Ml  =  M  =  4  — .499  =  3.501. 

M3  =  3.074  —  (—  .499)2  =  2.824999. 

Ms  =  -  1.^36  -  3(-  .499  X  3.074)  +  2(-  .499)3  =  2.417278. 

M4  =  24.284 -4(-.499x  -  1.936)  +  6(.249001  X  3.074)  -  3(-  499)*  = 

_  (2.417278)2  _    5.843232929 
Pl  ~  (2.S24999)3  ~  22.545241683 

24.826297  _  94. 826297 
^  ~  (&824999J*       ?.y»061935  "  3'110823- 

F  =  6  4-  3  X  0.259178  -  2  X  3.110823  =  -f  0.555888  (Type  I). 
6(3.11082  -  0.25918—  1) 


.55589 
—  21.9857 


=  19.9857. 


d  =  1.680774  X  .3111  =  .5230. 
d .  s=  .5230  X  19.9857  =  10.4519. 


b  =  .840387  4/16  X  20.9857  -f  0.25918  X  (21. 9857) 3  =  18.0448. 
18.0448  -  10.4519 


26  STATISTICAL   METHODS. 

a  a  =  18.0448  -  3.7965  =  14.2483. 

-,-i=%^  =  «s«. 

^.sa-jytsi.^ 

_    2000          (18.9846)  4/17.9846  g  171828'0883('°556  ~  *2643  ~~  *0704) 

~~  18.0448  y^  X  3.7840  X  14.2006 
=  475.24,    the  number  of  cases  in  the  modal  class. 
The  equation  of  the  theoretical  curve  is  thus 


where  x  is  the  difference  between  the  class  magnitude  and  the  mode, 
regarding  signs. 

Position  of  the  mode,    yQ  =  M  -  d  =  3.501  -  .523  ^  2.978. 
The  mean  percentage  deviation  of  the  theoretical  ordinates  from  the 
observed  ordinates  is  11.4£*  (Method  A).    This  is  calculated  as  follows: 


0.0 

—  6.1  40.7 
+  23.2  11.1 

—  30.1  8.2 
+    6.8  1.4 
4-    8.4  2.0 
+    4.9  1.8 

—  13.6  10.2 
-f    6.1  8.5 

-    2.1  9.5 

+    1.0  12.5 

+    0.4  20.0 


V 

f 

y 

observed 

theoretical 

•  1 

0 

0.0 

0. 

15 

21.1 

1 

209 

185.8 

2 

365 

395.1 

3 

482 

475.2 

4 

414 

405.6 

5 

277 

272.1 

6   . 

134 

147.6 

7 

72 

65.9 

8 

22 

24.1 

9 

8 

7.0 

10 

2 

1.6 

11 

0 

0.2 

12 

0 

0.0 

MULTIMODAL    CURVES. 

Multimodal  curves  are  given  when  the  frequency  in  the 
different  classes  exhibits  more  than  one  mode.  False  multi- 
modal  curves  result  from  too  few  observations,  or  when  the 
classes  are  made  too  numerous  for  the  variates.  By  increas- 
ing the  number  of  variates  or  by  making  the  classes  more 
inclusive  some  of  the  modes  disappear. 

*  The  mean  percentage  deviation  by  Duncker's  determination  with 
method  B  using  the  same  data  is  1.73#  of  area. 


THE  CLASSES  OF  FREQUENCY  POLYGOHS.    27 


4         5         6         T 
FIG.  24. 

Distribution  of  frequency  in  glands  of  swine, 
polygon  of  observed  frequency. 


8 


10 


—  •  —  -,  polygon  of  theoretical  frequency  (Type  I). 
-    -   -  -,  normal  frequency  polygon. 


38  STATISTICAL   METHODS. 

Multimodal  curves  differ  in  degree.  The  modes  may  be  so 
close  that  only  a  single  mode  (usually  in  an  asymmetrical 
curve)  appears  in  the  result;  or  one  of  the  modes  may  appear 
as  a  hump  on  the  other;  or  the  two  modes  may  even  be  far 
apart  and  separated  by  a  deep  sinus  (Figs.  25  to  28). 


\ 


N 


5.5    4.5    3.5    2.5    1.5    .5  0.5     1.5   2.5    3.5   4.5    5.5   6.5    75 
FIG.  25. 

Pearson  has  offered  a  means  of  breaking  up  a  compound 
curve  with  apparently  only  one  mode  into  two  curves  having 
distinct  modes;  but  this  method  is  very  tedious  and  rarely 
applicable. 


o 

FIG.  26. 

The  index  of  divergence  of  two  modes  of  a  multi- 
modal  curve  is  the  distance  between  the  modes  expressed  in 


THE  CLASSES  OF  FREQUENCY  POLYGONS.    29 


terms  of  the  standard  deviation  of  the  more  variable  of  the 
components.* 

The  index  of  isolation  of  two  masses  of  variates 
grouped  about  adjacent  modes  is  the  ratio  of  the  depression 
between  the  modes  to  the  height  of  the  shorter  mode. 

The  meaning  of  multimodal  curves  is  diverse.     Sometimes 


\ 


FIG.   27. 

they  indicate  a  polymorphic  condition  of  the  species,  the  modes 
representing  the  different  type  forms.     This  is  the  case  with 


3210123 

01234 

FIG.  28. 

the  number  of  ray  flowers  of  the  white  daisy  which  has  modes 
at  8,  13,  21,  34,  etc.  Sometimes  they  indicate  a  splitting  of  a 
species  into  two  or  more  varieties. 

*  I  have  proposed  (Science,  VII,  685)  to  measure  the  divergence  in  a 
unit  =  3  X  Standard  Deviation,  which  has  certain  advantage  in  species 
study. 


30  STATISTICAL    METHODS. 


CHAPTER   IV. 
CORRELATED  VARIABILITY. 

Correlated  variation  is  such  a  relation  between  the  magni- 
tudes of  two  or  more  characters  that  any  abmodality  of  the 
one  is  accompained  by  a  corresponding  abmodality  of  the 
other  or  others. 

The  methods  of  measuring  correlation  depend  upon  the 
assumption  that  the  variates  of  the  characters  compared  are 
distributed  normally  about  the  mode.  The  method  is  approxi- 
mately applicable  to  cases  where  the  distribution  of  variates  is 
slightly  skew. 

TThe  principles  upon  which  the  measure  of  correlated,  varia- 
tion rest  are  these.  When  we  take  individuals  at  random  we 
find  that  the  mean  magnitude  of  any  character  is  equal  to  the 
mean  magnitude  of  this  character  in  the  whole  population. 
Deviation  from  the  mean  of  the  whole  population  in  any  lot  of 
individuals  implies  a  selection.  If  we  select  individuals  on  the 
basis  of  one  character  (A,  called  the  subject}  we  select  also  any 
closely  correlated  character  (B,  called  the  relative)  (e.g.  leg- 
length  and  stature).  If  perfectly  correlated,  the  index  of 
abmodality  of  B  will  be  as  great  as  that  of  A  or 
Index  abmodality  of  relative  ___  .. 
Index  abmodality  of  subject 

If  there  is  no  correlation,  then  whatever  the  value  of  the 
index  of  modality  of  the  subject,  that  of  the  relative  will  be 
zero  and  the  coefficient  of  correlation  will  be 

Index  of  abmodality  of  relative  __    0    _  . 

Index  of  abmodality  of  subject        m 

The  coefficient  of  correlation  is  represented  in  formulas  by 
the  letter  p.  We  cannot  find  the  degree  of  correlation  between 
two  organs  by  measuring  a  single  pair  only  ;  it  is  the  correla- 
tion "  in  the  long  run  "  which  we  must  consider.  Hence  we 
must  deal  with  masses  and  with  averages. 


CORKELATED    VARIABILITY. 


31 


5s    5" 

0          g 

g$g8£S38£i2£ 

t>  Q 

0)        . 

Q  CO 

C*i-iOOOOr-lCJ<MCOCO 
1             1             1             1 

?!   "2 

O«3OOOO*^OT-(C5O 

I«l*tC<MS{i»v*i5fc»£4.ej 

V     . 
Q   20 

7  7    i  *? 

">  S 

OOTfCOOOT}«O--<OCOO 

Tj«OOCO-^r}<T}<l-O5-305S 

osi-is$coco<?*os«oosr-T}< 

®  0 

fi^ 

ci      ci      ~       !              ^      ~      <* 

M 

O-u? 

oj«t-i 

u  ® 

g~ 

O-T-iOlCOOQ^OtOtOt—         C5 

a 

^ 

O 

os 

to 

O         iT2         O*         OJ        »-i 

OD 

<Q 

* 

«-«•-«-        ! 

t« 

O 

^ 

CO 

-    s    S    s    -    -     : 

(0 

O 

oi 

»-itoeoc?ooQO»nco 

(M        O        TP        ^i 

O 

to 
* 

J>CX3l--T-«OOOCO'-'            • 
W        t-        0        0        W 

^ 

O 
O 

-  s  §  a  a  s  -  -    ;    j 

CO 

iO 

o" 

1 

•    S    g    S    X    -    -     1     :     I 

s 

c<      c»      rf      QO      t^      t-         

Of 

7 

-    *    *           :      :     :     :     : 

s 

! 

m 

00          T»<          (N              '"'*'''* 

CO 

1 

bX) 
O 

cc 

•* 
*3 

eg£iC1O^5i^'^<'^''^''T*'^<'^1'T*( 

JS 

J3 

*-i 

1 

^ 

'>dcoO'-iOOi-<o*co'^<ioto 
®  5      I       I       I       i 
p  8 

«M 

to 

0) 

s 

Deviations 

«M 

Jl 

e8  be 

32  STATISTICAL   METHODS. 

In  studying  correlation  one  (either  one)  of  the  characters  is 
regarded  as  subject  and  the  other  as  relative.  A  correlation 
table  is  then  arranged  as  in  the  example  on  page  29,  which 
gives  data  for  determining  the  correlation  between  the  num- 
ber of  Miillerian  glands  on  the  right  (subject)  and  left  (rela- 
tive) legs  of  male  swine. 

METHODS  OF  DETERMINING  COEFFICIENT  OF  CORRELATION. 

Gallon's  graphic  method.  On  co-ordinate  paper 
draw  perpendicular  axes  X  and  T\  locate  a  series  of  points 
from  the  pairs  of  indices  of  abmodality  of  the  relative  and  sub- 
ject corresponding  to  each  subject  class.  The  indices  of  the 
subjects  are  laid  off  as  abscissas  ;  the  indices  of  the  relatives 
as  ordinates,  regarding  signs.  Get  another  set  of  points  by  mak- 
ing a  second  correlation  table,  regarding  character  B  as  subject 
and  character  A  as  relative.  Then  draw  a  straight  line  through 
these  points  so  as  to  divide  the  region  occupied  by  them  into 
halves.  The  tangent  of  the  angle  made  by  the  last  line  with 
the  horizontal  axis  XX  (any  distance  yp,  divided  by  xp)  is  the 
index  of  correlation. 

A  more  precise  method  is  given  by  Pearson  as  follows: 
Sum  of  products  (deviation  subj.  class  X  deviation  each  assoc. 

rel.  class  X  no.  of  cases  in  both) 

total  no.  oTludivs.  X  Stand.  Dev.   of  subject  x  Stand.  Dev. 
of  relative  ; 

or,  expressed  in  a  formula  : 

^(dev.  x  X  dev.  y  X  /) 

p  —  

ncri<Tz 

This  method  requires  finding  many  products  in  the  numera- 
tor, as  many  sets  of  products  as  there  are  entries  in  the  body  of 
the  correlation  table.  A  portion  of  the  products  to  be  found 
is  indicated  below  ; 

(-  3.540  X  8 

-  3.547  X   4-  2.540  X  5 

(-  1.540  X  2 

f_  3.540  X      4 
|  -  2.540  x  161 

-  2.547  XX-  1-540  x    58 

|  -  0.540  X      9 
t-  0.460  X      3 
etc. 


CORRELATED    VARIABILITY.  33 

A  brief  method  of  finding  p  is  given  by  Duncker  as  follows: 


.  . 

p  is  composed  of  two  factors:  —  —  and  - 

n  a  i  er2 

To  find  ^(dev.  K  X  clev.  y  x  f\ 

n 

Separate  the  deviation  from  the  mean  of  each  class  into  its 
integral  and  its  fractional  parts  ;  the  fractional  parts  for  all 
classes  below  the  mean  will  be  equal  to  the  fractional  part  of 
the  mean  ;  of  all  classes  above  the  mean,  to  the  complement 
of  that  number.  Designate  the  integral  parts  of  the  variants 
of  the  subject  by  ±  Xi  ;  of  the  relatives  by  ±  X2,  and  the  frac- 
tional complement  parts  of  the  means  of  subject  or  relative  by 
&,  ?2.  Let  /equal  the  frequency  of  any  deviation  in  the  com- 
bination X,X.2,  as  shown  in  the  correlation  table.  Draw  rect- 
angular co-ordinates  as  shown  on  page  34  through  the  zero- 
point  of  the  correlation  table.  Number  the  N.  "W.  quadrant, 
which  should  include  negative  deviations  of  both  subject  and 
relative  variants,  I  ;  the  N.  E.  quadrant,  II  ;  the  S.  W. 
quadrant  containing  solely  positive  deviations  III  ;  and  the 
S.  E.  quadrant,  IV.  Then  if  2t,  2llt  etc.,  indicate  a  summa- 
tion for  the  quadrant  I,  II,  etc.,  and  having  regard  to  signs  ; 

-  2I(fXl)  -  3 


The  numerator  of  this  fraction  consists  entirely  of  whole 
numbers  ;  of  them  the  following  are  on  their  own  account      / 


positive:     2z(fXiX>),   2nlfX1Xj,    2i(f),   2n(fXJ, 

SjuC/ 
negative  :     2V(fXlX*),  Sn^fX.X^  ^i(fX,},  2j(fX9). 


Rule  :  (1)  Find  products  of  integral  parts  of  deviations  of 
both  subject  and  relative  and  the  combination  frequency,  for 
all  four  quadrants,  and  take  their  sum. 

(2)  Subtract  successively  the  sum  of  the  products  of  the  sub- 
ject deviations  in  the  first  quadrant  multiplied  by  the  fre- 
quency, and  the  sum  of  the  products  of  the  relative  deviations 


34 


STATISTICAL    METHODS. 


in  the  first  quadrant  multiplied  by  the  frequency.     Since  these 
are  negative  values  they  will  be  actually  added. 

(3)  Add  the  suin  of  the  numbers  in  the  first  quadrant  ;  sub. 
tract  the  sum  of  the  products  of  the  integral  parts  of  the  rel- 
ative deviations  by  the  frequency  in  the   second   quadrant  ; 
subtract  the  sum  of  the  products  of  the  subject  deviations  of 

he  third  quadrant  multiplied  by  their  frequency. 

(4)  Divide  the  algebraic  sum  of  (1),  (2),  and  (3)  by  the  number 
of  variates,  and  from  the  quotient  subtract  the  product  of  the 
complement-fractional  parts  of  the  mean  value  of  the  subject 
and  relative. 


To  get  p,  divide  -   —  —  by  the  product  of 


and  cr2. 


The  probable  error  of  the  determination  of  p  is 
0.6745(1  -  p^ 


yn(l  -f  p) 

Example.     Correlation  in  number  of  Mullerian  glands  on 
right  and  left  legs  of  2000  male  swine. 

Mean,  right  leg,  =  3.5465  ;        Mean,  left  leg,  =  3.5395 
o-,  =  1.7195;  <ra  =  1.7304 

Right  leg,  subject:  Left  leg,  relative. 


X, 

-  3 

-2 

-1 

0 

0 

1 

2 

3 

4 

5 

6 

Rel. 

class  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Sub. 

class  (I) 

(ID 

0  - 

-3   8 

5 

2 

1  - 

-2   4 

151 

'58 

9 

3 

2  - 

-  1   2 

65 

154 

96 

28 

7 

1 

3 

0 

14 

88 

173 

128 

28 

6 

4 

0 

5 

27 

119 

153 

77 

26 

3 

1 

5 

1 

1 

7 

24 

92 

101 

52 

11 

9 

6 

2 

8 

16 

58 

48 

16 

7 

2 

7 

3 

1 

8 

20 

18 

17 

9 

5 

8 

4 

1 

3 

5 

3 

2 

2 

9 

5 

1 

3 

3 

2 

2 

1 

10 

6  (III) 

1 

(IV) 

CORRELATED    VARIABILITY.  35 

*)  =  1142-9—  9+1652  ^ 
2(fX,)  +  Sr(f)  5125 

=  4-  806  +  814  +  829   ^   -*-  »  =  -          =  2.5625 
.)  -  2m(fXl) 

=  _  49  -  51  J 


-  S,«,  =  -4535  X.  4605= 


=  1.7195  X  1.7304  =  2.9754;  p  =  =  .7919 


1/2000  X  1.627 

SPURIOUS  CORRELATION  IN  INDICES. 

When  two  characters  A  and  B  are  measured  in  each  individual 
ofaseries  of  individuals,  and  each  absolute  magnitude  is  trans- 
formed into  an  index  by  dividing  it  by  the  magnitude  of  a 
third  character  C  as  found  in  the  same  individual,  a  spurious 
correlation  will  be  found  to  exist  between  the  indices  of 
A  .  B 

c  and  c 

Let  Vi   =  the  coefficient  of  variation  of  A  ; 

oa  =    ••  »          •«         «          «  B; 

PO  —    "  "  spurious  correlation. 

The  precise  method  of  using  pQ  in  modifying  any  determina- 
tion of  p  is  uncertain.  Pearson  recommends  using  p  —  p0  as 
the  true  measure  of  "organic  correlation"  in  the  case  of 
indices. 

HEREDITY. 

Heredity  is  a  certain  degree  of  correlation  between  the 
abmodality  of  parent  and  offspring.  The  statistical  laws  of 
heredity  deal  not  with  relations  between  one  descendant  and 


36  STATISTICAL  METHODS. 

its  parent  or  parents,  but  only  with  mean  progeny  of  mean 
parents.  Any  group  of  selected  parents  is  called  a  parentage, 
the  progeny  of  a  parentage  is  called  a  fraternity. 

In  uniparental  inheritance,  as  in  budding  or  asexual 
generation,  heredity  of  any  character  is  measured  by  the  coef- 
ficient of  correlation  between  the  abmodality  in  a  parentage 
and  the  abmodality  |of  the  corresponding  fraternity.  More 
strictly,  since  the  variability  of  the  character  in  the  second 
generation,  crz,  may  (as  a  result  of  selection  or  of  environ- 
mental change)  be  different  from  the  variability  of  the  char- 
acter in  the  first  generation,  crly  the  index  should  be  taken  as 


The  probable    error   of  this  determination  is 

-  —  ,  in  which  Pi2  means  the  correlation  coeffi- 
n 

cieDt  between  the  filial  character  and  that  of  the  single  parent 
under  consideration. 

The  variability  of  the  fraternity  is  to  variability  of  offspring 
in  general  as  |/1  —  p2  is  to  1. 

In  biparental  inheritance,  if  there  is  no  evidence  of 
assortative  mating,  or  correlation  between  the  two  parents  in 
the  character  in  question,  the  mean  abmodality  of  any  frater- 
nity will  be 


cr,_  cr, 

hi  =  p3  —  7ia  +  PI  —  /i*, 
cr2  cr3 

where  h\   =  average  abmodality  of  fraternity  ; 

7*2  —  average  abmodality  of  male  parent  ; 

hs  —  average  abmodality  of  female  parent  ; 

pa  =  correlation     coefficient    between    fraternity    and 

female  parent  ; 
p8  =  correlation  coefficient  between  fraternity  and  male 

parent  ; 

o-!  =  standard  deviation  of  fraternity  ; 
cr2  =  standard  deviation  of  male  parent  ; 
cr3  =  standard  deviation  of  female  parent. 


CORRELATED    VARIABILITY.  37 

When   assortative   mating   occurs,   as  is  usually  case,   the 
abmodality  of  a  fraternity  is  given  by 

7 

7"= 


where  p\  =  correlation  between   male   and  female  parents. 
The  other  letters  have  the  same  signification  as  before. 

The  strength  of  heredity  in  assortative  mating  is  measured 
by  the  formula 


1  -  /V    '  crV 

Galton  ('97)  has  shown  that  an  individual  inherits  not  only 
from  his  parents,  but  also  from  his  grandparents,  great-grand- 
parents, and  so  on.  The  heritage  from  his  2  parents  together 
is,  on  the  average,  50$  or  J  of  the  whole  ;  from  the  4  grand- 
parents 25$  or  J  ;  from  the  8  great-grandparents  12.5$  or  J  ; 

from  the  Tith  ancestral  generation  —  of  the  whole  ;  the  total 

heritage  adding  up  100$.  This  law  has  been  generalized  by 
Pearson  ('98)  as  follows  : 

1  O*OT      .    1  CTO.         1  (TO.          1   CTO 

li\  =  ~  --  ki  +  i  --  &a  +  ~  --  &3  H  ---  #4  +  •  • 

2  <rl      ^  4  o-a      ^  8  cr3       ^  16  <74 

where  A!   =  average  abmodality  of  fraternity. 
<T0  =  standard  deviation  of  fraternity. 
(7i,  cra  .  .  .  <rs  =  standard  deviation  of  mid-parent  of 
1st,  2d  ...  5th  ancestral  generation. 
ki  =  abmodality  of  mid-parent  of  1st  ancestral  genera- 

tion. 
&3,  &3  .  .  .  ks  =  abmodality  of  mid-parent  of  2d,    3d 

.  .  .  5th  ancestral  generation. 

The  abmodality  of  the  mid-parent  of  any  degree  of  ancestry 
may  be  taken  as  the  average  abmodality  of  all  the  contributory 
ancestors  of  that  generation. 


38  STATISTICAL   METHODS. 


CHAPTEK  Y. 
SOME  APPLICATIONS  OF  STATISTICAL  BIOLOGICAL  STUDY. 

The  Laws  of  Variation.  Darwin  and  others  have 
formulated  certain  laws  of  variation,  such  as  the  law  that 
specific  characters  are  more  variable  than  generic  ones  ;  that 
highly  aberrant  characters  are  more  variable  than  more  usual 
ones  ;  that  males  are  more  variable  than  females.  These  laws 
can  be  established  only  by  a  determination  of  the  Index  or 
Coefficient  of  Variation  in  critical  cases. 

The  causes  of  variation  can  be  determined  only  by  r 
quantitative  study  of  the  relation  between  specific  change  and 
environmental  change,  or  a  knowledge  of  the  degree  and  fre- 
quency of  sports. 

The  effect  of  selection  in  causing  a  greater  death  rate  on 
one  side  of  the  mean  than  on  the  other  side — the  production  of 
skewness — requires  the  quantitative  method  for  its  complete 
study.  The  change  in  the  mode  and  in  the  index  of  skewness 
measures  the  progress  of  the  effect  of  selection. 

The  origin  of  species  through  geographical  segrega- 
tion can  be  studied  by  the  determination  of  place-modes  ;  that 
is,  the  modal  condition  of  specific  characters  of  one  and  the 
same  species  in  various  localities.  The  progress  of  specific 
differentiation  will  be  measured  by  the  change  in  place-modes 
from  decade  to  decade,  or  by  the  formation  of  a  binomial  curve 
in  the  place  of  a  modal  one  ;  and  by  the  gradual  separation  of 
the  two  modes  of  a  binomial  curve. 

The  definition  of  species  may  be  improved  by  being 
rendered  more  quantitative.  The  relative  importance  of  the 
various  criteria  used  in  separating  species  may  be  determined 
by  finding  that  character  in  which  there  is  least  intergrading 
between  the  modal  condition  characteristic  of  the  two  races. 
Thus  if  for  two  species  or  varieties  of  birds  both  total  length 
and  form  of  bill  show  two  modes,  the  better  criterion  is  that 
in  which  the  modes  are  farthest  apart  or  in  which  the  inter- 
grades  are  fewest. 


STATISTICAL    BIOLOGICAL    STUDY.  39 

A  basis  for  an  arbitrary  distinction  between  species 

and  varieties  may  be  gained  by  determining  a  degree  of 
divergence  and  of  isolation  which  shall  be  used  to  distinguish 
the  two.  A  degree  of  divergence  of  thrice  the  standard  devia- 
tion has  been  suggested  as  a  convenient  line  between  species 
and  varieties. 

Quantitative  studies  in  correlation  will  give  us  new  cri- 
teria for  homology  by  telling  us  the  relative  morphoge- 
netic  kinship  of  the  parts  of  the  body. 

Quantitative  studies  in  heredity  will  give  definitive  informa- 
tion on  prepotency  of  sex  or  race.  By  examining  hybrids 
quantitatively  and  comparing  them  with  their  parents  we  shall 
unravel  the  laws  of  inheritance  in  cross-breeding  and  the  prin- 
ciples of  mixing  characters  in  biparental  inheritance. 

In  a  word,  by  the  use  of  the  quantitative  method  biology 
will  pass  from  the  field  of  the  speculative  sciences  to  that  of 
the  exact  sciences. 


40  STATISTICAL    METHODS. 

SELECTED  BIBLIOGRAPHY 

OF  WORKS   ON    THE    QUANTITATIVE  STUDY   OF   ORGANISMS. 

AMANN,  J.,  '96.  Application  du  calcul  des  probability's  & 
1'etude  de  la  variation  d'un  type  vegetal.  Bull,  de 
1'Herb.  Bossier.  Geneve  et  Bale.  IV,  578-590. 

BREWSTER.  E.  T.,  '97.  A  Measure  of  Variability  and  the 
Relation  of  Individual  Variations  to  Specific  Differences. 
Proc.  Amer.  Acad,  Arts  and  Sci.,  XXXII,  268-280. 

BUMPUS,  H.  C.,  '97.  The  Variations  and  Mutations  of  the 
Introduced  Sparrow.  Biol.  Lect.  Woods  Holl,  1896,  1-15. 

BUMPUS,  H.  C.,  '98,  The  Variations  and  Mutations  of  the 
Introduced  Littorina.  Zool.  Bull.,  I,  247-259. 

DAVENPORT,  C.  B.,  and  J.  W.  BLANKINSHIP,  '98.  A  Precise 
Criterion  of  Species.  Science,  VII,  685-695. 

DAVENPORT,  C.  B.,  and  C.  BULLARD,  96.  Studies  in  Morpho- 
genesis, VI.  A  Contribution  to  the  Quantitative  Study 
of  Correlated  Variation  and  the  Comparative  Variability 
of  the  Sexes.  Proc.  Amer.  Acad.  Arts  and  Sci. ,  XXXII, 
85-97. 

DUNCKER,  G.,  '97.  Correlation  Studieu  an  den  Strahlzahlen 
einiger  Flossen  von  Acerina  cernua~L.  Biol.  Centralbl., 
XVII,  785-794  ;  815-831. 

DUNCKER,  G.,  '98.  Bemerkung  zu  dem  Aufsatz  von  H.  C. 
Bumpus  "  The  Variations  and  Mutations  of  the  Introduced 
Littorina."  Biol.  Centralbl.,  XVIII,  569-573. 

DUNCKER,  G.,  '99.  Die  Methode  der  Variation  s-Statistik. 
Arch.  f.  Entwickelungs-Mechan.  d.  Organismen,  VIII, 
112-183.  [The  most  important  elementary  presentation 
of  the  subject ;  extensive,  nearly  complete  bibliography.] 

EIGENMANN,  C.  H.,  '95.  Leuciscus  balteatus  (Richardson), 
a  Study  in  Variation.  Amer.  Naturalist,  XXIX,  10-25, 
Pis.  1-5. 

EIGENMANN,  C.  H.,  '96.  The  Study  of  Variation.  Proc. 
Indiana  Acad.  Sci.,  V,  265-278.  [Extensive  bibliography.] 

FECHNER,  G.  T.,'97.  Kollektivmasslehre.  Im  Auftrage  der 
Koniglich  Sachsischen  Gesellschaft  der  Wissenschaften 
herausgegeben  von  Gottl.  Friedr.  Lipps.  Leipzig  :  Engel 
mann.  483pp.  [Important  but  too  much  neglected  work.] 


SELECTED    BIBLIOGRAPHY.  41 

FIELD,  W.  L.  W.,  '98.     A  Contribution  to  the  Study  of  Indi- 
vidual  Variation  in  the  Wings   of   Lepidoptera.      Proc. 

Amer.  Acad.  Arts  and  Sci. ,  XXXIII,  389-395. 
GALTON,  F.,'88.    Correlations  and  their  Measurement,  chiefly 

from  Anthropometric  Data.     Proc.    Roy.   Soc.  London, 

XLV,  136-145. 

GALTON,  F.,  '89.     Natural  Inheritance.    London  :  Macmillan. 
GALTON,  F.  '97.     The  Average  Contribution  of  each  several 

Ancestor  to  the  total  Heritage  of  the  Offspring.     Proc. 

Roy.  Soc.  London,  LXI,  401-413. 
LUCAS,  F.  C.,'98.     Variation  in  the  Number  of  Ray-flowers  in 

theWhiteDaisy.  Amer.  Naturalist,  XXXII,  509-511.  2figs. 
LUDWIG,  F. ,   '95.     Ueber  Variationskurven  uud  Variations- 

flachen  der  Pflanzen.   Bot.  Centralbl.,  LXIV,  1-8  et  folg. 

2  Tafn. 
LUDWIG,  F.,  '96.     Weiteres  iiber  Fibonacci-Kurven  und  die 

numerische  Variation  der  gesammten  Bliithenstande  der 

Kompositen.     Bot.  Centralbl.  LXVIII,  1  et  folg.     1  Taf. 
LUDWIG,  F.,  '96.     Eine  ftinfgipfplige  Variations-Kurve.    Ber. 

deutsch.  Bot.  Ges.,  XIV,  204-207.     1  fig. 
LUDWIG,  F.,  '98.    Die  pflanzlichen  Variation  s-Kurven  und  die 

Gauss 'sche    Wahrscheinlichkeitskurve.     Bot.  Centralbl., 

LXXIII,  241-250  et  folg.    1  Taf. 
LUDWIG,  F.,  '98.     Ueber  Variationskurven.    Bot.  Centralbl., 

LXXV,  97-107  ;  178-183.     1  Taf. 
MOENKHAUS,  W.  J.,  '96.     The  Variation  of  Etheostoma  cap- 

rodes  Rafinesque  in  Turkey  lake  and  Tippecanoe  lake. 

Proc.  Indiana  Acad.  Sci.,  V.,  278-296. 
PEARSON,  K.,  '94.     Contributions  to  the  Mathematical  Theory 

of    Evolution.     [I.    On    the   Dissection    of    Frequency 

Curves.]      Phil.  Trans.   Roy.  Soc.    London,  CLXXXV, 

A,  71-110.     Pis.  1-5. 
PEARSON,  K.,  '95.     Contributions,  etc.,  II.     Skew  Variation 

in    Homogeneous     Material.    Phil.     Trans.    Roy.    Soc. 

London,  CLXXXVI,  A,  343-414.     10  Pis. 
PEARSON,  K.,  '96.     Mathematical  Contributions  to  the  Theory 

of  Evolution,  III.     Regression,  Heredity,  and  Panmixia. 

Phil.  Trans.  Roy.  Soc.   London,  CLXXXVII,  A,  253-318. 
PEARSON,  K.,  '97.     Mathematical  Contributions,  etc.     On  a 

Form  of  Spurious  Correlation,  which  may  Arise  when 


42  STATISTICAL   METHODS. 

Indices  are  used  in  the  measurement  of  Organs.  Proc. 
Roy.  Soc.  London,  LX,  489-498. 

PEARSON,  K.  '98.  Mathematical  Contributions,  etc.  On  the 
Law  of  Ancestral  Heredity.  Proc.  Roy.  Soc.  London, 
LXII,  386-412. 

PEARSON,  K.,  and  L.  K  G.  FILON,  '98.  Mathematical  Con- 
tributions, etc.,  IV.  On  the  Probable  Errors  of  Frequency 
Constants  and  on  the  Influence  of  Random  Selection  on 
Variation  and  Correlation.  Phil.  Trans.  Roy.  Soc. 
London,  CXCI,  A,  229-311. 

THOMPSON,  H.,  '94.  On  Correlations  of  Certain  External 
Parts  of  Palaemon  serratus.  Proc.  Roy.  Soc.  London, 
LV,  234-240. 

VERSCHAFFELT,  E.,  '95.  Ueber  Asymmetrische  Variations- 
kurven.  Ber.  deutsch.  Bot.  Ges.,  XIII,  348-356.  1  Taf. 

DE  VRIES,  H.,  '94.  Ueber  halbe  Galton-Kurven  als  Zeichnen 
diskontinuirlichen  Variation.  Ber.  deutsch.  Bot.  Ges., 
XII,  197-207.  Taf.  X. 

DE  VRIES,  H.,  '95.  Eine  zweigipfelige  Variations-Kurve. 
Arch.  f.  Entwickelungsmechanik,  II.  52-65.  1  Taf . 

WARREN,  E.,  '96.  Variation  in  Portunus  depurator.  Proc. 
Roy.  Soc.  London,  LX,  221-243. 

WARREN,  E.,  '97.  An  Investigation  on  the  Variability  of  the 
Human  Skeleton  with  Especial  Reference  to  the  Naquada 
Race.  Phil.  Trans.  Roy.  Soc.  London,  CLXXXIX,  B, 
135-227.  PI.  22. 

WELDON,  W.  F.  R.,  '90.  The  Variations  occurring  in  Certain 
Decapod  Crustacea,  I :  Crangon  milgaris.  Proc.  Roy. 
Soc.  London,  XLVII,  445-453. 

WELDON,  W.  F.  R.,  '92.  Certain  Correlated  Variations  in 
Crangon  vulgaris.  Proc.  Roy.  Soc.  London,  LI,  2-21. 

WELDON,  W.  F.  R.,  '93.  On  Certain  Correlated  Variations  in 
Carcinus  maenas.  Proc.  Roy.  Soc.  London,  LIV,  318- 
329. 

WELDON,  W.  F.  R.,  '95.  Report  of  the  Committee  for  Con- 
ducting Statistical  Inquiries  into  the  measurable  Char- 
acteristics of  Plants  and  Animals.  Part  I  :  An  Attempt 
to  Measure  the  Death-rate  due  to  Selective  Destruction  of 
Car  emus  maenas  with  respect  to  a  Particular  Dimension. 
Proc.  Roy.  Soc.  London,  LVII,  360-379. 


EXPLANATION    OF   TABLES.  43 


EXPLANATION  OF  TABLES. 

I.  Formulas.  In  this  table  the  principal  formulas  used 
in  the  calculation  of  curves  are  brought  together  for  conven- 
ient reference.     The  meanings  of  the  letters  are  explained  in 
the  text. 

II.  Certain  constants   and    their  logarithms. 

This  table  includes  the  constants  most  frequently  employed 
in  the  calculations  of  this  book. 

III.  Table  of  ordinates  of  normal  curve.    This 
table  is  for  comparison  of  a  normal  frequency  polygon  con- 
sisting of  weighted  ordinates  with  the  theoretical  curve. 

Example:    M  =  14.157  ;         a  =  0.604  ;         ly<>  =  1255. 
(See  page  19.) 

Entries  in  Table 
V  -  M   corresponding  to 
V  V-M —  V-  M  2/o  y  f 

a 

11  -  3.157   5.2     .000004    X  1255  =   0.0    1 

12  -  2.157   3.6     .0015     X  1255  =   1.8    2 

13  - 1.157   1.9     .164      X  1255  =  20108  189 

IV.  Table   of  values  of  probability   integral. 

This  table  is  for  comparison  of  a  normal  frequency  polygon 
consisting  of  rectangles  with  the  theoretical  curve. 
Example:  M.  5.24  ;  cr  =  0.987.     (See  page  12). 


' 

• 

Class 
limits 

Deviation 
from 

X 
<T 

x  -+•  <r 

1 

4 

3.0 

—2.24 

—  2.27 

1 

4 

3.5 

—  1.74 

—  1.76 

46.1  

3 

12 

4.0 

-1.24 

-1.26 

39.6  "* 

3 

7 

12 
28 

4.5 
5.0 

-0.74 
-0.24 

-0.75 
-0.25 

27.3  1 
The     oret 
9.9)20.5    55.2 

[       Y 

ical 
79.6 

- 

fre 
92.4 

• 

quency 
97.7 

5 

20 

5.5 

0.26 

0.27 

10.6J2<?     1  bo 
Ob  |  serv 

84 
ed 

fre 

IOO 

quency 

3 

12 

6.0 

0.76 

0.77 

27.9  J 

6.5 

1.26 

1.28 

40.0  

1 

4 

7.0 

1.76 

1.78 

46  3  

1 

4 

— 



7.5 

2.26 

2.29 

48  9 

25 

100 

44  STATISTICAL   METHODS. 

In  the  example,  the  curve  of  which  is  shown  in  Fig.  22,  the 
frequency  between  the  limits  is  given  in  column  /;  the  fre- 
quency reduced  to  percents  in  column  headed  %.  The  —  of 

the  limit  is  found  and  the  entries  in  Table  IV  corresponding 
to  the  quotient  are  taken.  These  are  added  in  pairs  as  indi- 
cated, one  above  and  one  below  the  mean,  and  the  sum  is 
compared  with  the  sum  of  the  observed  cases  within  those 
limits  (in  italic  figures).  The  closeness  of  agreement  indicates 
the  closeness  with  which  the  observed  frequency  follows  the 
normal  frequency. 

V.  Table  of  log  T  functions  of  q.  This  table 
will  enable  one  to  solve  the  equations  for  yQ  given  on  page  23. 
The  table  gives  the  logarithms  of  the  values  of  F  functions 
only  within  the  range  p  =  1  to  2.  As  all  values  of  the  func- 
tion within  these  limits  are  less  than  1,  the  mantissa  of  the 
logarithms  is  —  1;  but  it  is  given  in  the  table  as  10  —  1  =  9, 
as  is  usually  done  in  logarithmic  tables. 

Supposing  the  quantity  of  which  we  wish  to  find  the  value 
reduced  to  the  form  r(4.273).  The  value  cannot  be  found 
directly  because  the  value  of  p  is  larger  than  the  numbers  in 
the  table  (1  to  2).  The  solution  is  made  by  aid  of  the  equation 
r(p  +  l)=pr(p),  thus: 

log  r(1.273)  =  9.955185 
log     1.273  =  0.104828" 

log  T(2. 273)  =  0.060013 
log      2.273  =0.356599 

,      log  T(3.273)  =  0.416612 
log     3.273  =0.514946 

log  T(4.273)  =  0.931558 

or,  more  briefly,  logP(1.273)  =  9.955185 
log  1.273  =  .104828 
log  2.273  =  .356599 
log  3.273  =  .514946 

log  r(4.273)  =  0.931558  =  log  8.542 


EXPLANATION   OF   TABLES.  45 

VI.  Table  of  reduction  from  the  common  to 
the  metric  system.    This  is  given  first  for  whole  inches 
from  1  to  99  excepting  even  tens,  which  may  be  got  from  the 
first  line  of  figures  by  shifting  the  decimal  point  one  place 
to  the  right.     The  table  may  be  used  for  hundredths  of  an 
inch  by  shifting  the  decimal  point  two  places  to  the  left. 
Other  fractions  than  decimals  are  given  in  the  lower  tables. 

VII.  First  to  sixth  powers  of  integers  from 

1  to  3O.     This  table  is  useful  in  calculating  moments. 

VIII.  Squares,   cubes,   square  roots,   and  re- 
ciprocals of  numbers  from  1  to  1O54.     The  use 

of  this  table  can  be  extended  by  using  the  principle  that  if  auy 
number  be  multiplied  by  n,  its  square  is  multiplied  by  n*,  its 

cube  by  ns,  and  its  reciprocal  by  — . 

IX.  Logarithms    of  numbers    to    six   places. 

The  following  explanation  of  the  use  of  the  logarithmic  tables 
is  taken  from  Seaiies'  Field  Engineering,  pp.  257-263  [ed. 
1887]. 

APPENDIX  IX.— The  logarithm  of  a  number  consists  of 
two  parts,  a  whole  number  called  the  characteristic,  and  a  deci- 
mal called  the  mantissa.  All  numbers  which  consist  of  the 
same  figures  standing  in  the  same  order  have  the  same  man- 
tissa, regardless  of  the  position  of  the  decimal  point  in  the 
number,  or  of  the  number  of  ciphers  which  precede  or  follow 
the  significant  figures  of  the  number.  The  value  of  the  char- 
acteristic depends  entirely  on  the  position  of  the  decimal  point 
ia  the  number,  It  is  always  one  less  than  the  number  of 
figures  in  the  number  to  the  left  of  the  decimal  point.  The 
value  is  therefore  diminished  by  one  every  time  the  decimal 
point  of  the  number  is  removed  one  place  to  the  left,  and  vice 
versa.  Thus 

Number.  Logarithm. 

13840.  4.141136 

1384.0  3.141136 

138.40  2.141136 

13.84  1.141136 

1.384  0.141136 

.1384  —1.141136 

.01384  —2.141136 

.001384  —3.141136 

etc.  etc. 


46 


STATISTICAL   METHODS. 


The  mantissa  is  always  positive  even  when  the  characteristic 
is  negative.  We  may  avoid  the  use  of  a  negative  characteristic 
by  arbitrarily  adding  10,  which  may  be  neglected  at  the  closf 
of  the  calculation.  By  this  rule  we  have 

Number.  Logarithm. 

1.384  0.141136 

.1384  9.141136 

.01384  8.141136 

.001384  7.141136 

etc,  etc. 

No  confusion  need  arise  from  this  method  in  finding  a  number 
from  its  logarithm;  for  although  the  logarithm  6.141136  repre- 
sents either  the  number  1,384,000,  or  the  decimal  .0001384,  yet 
these  are  so  diverse  in  their  values  that  we  can  never  be  uncer- 
tain in  a  given  problem  which  to  adopt. 

The  table  IX.  contains  the  mantissas  of  logarithms,  car- 
ried to  six  places  of  decimals,  for  numbers  between  1  and  9999, 
inclusive.  The  first  three  figures  of  a  number  are  given  in  the 
first  column,  the  fourth  at  the  top  of  the  other  columns.  The 
first  two  figures  of  the  mantissa  are  given  only  in  the  second 
column,  but  these  are  understood  to  apply  to  the  remaining 
four  figures  in  either  column  following,  which  are  comprised 
between  the  same  horizontal  lines  with  the  two. 

If  a  number  (after  cutting  off  the  ciphers  at  either  end)  con- 
sists of  not  more  than  four  figures,  the  mantissa  may  be  taken 
direct  from  the  table ;  but  by  interpolation  the  logarithm  of  a 
number  having  six  figures  may  be  obtained.  The  last  column 
contains  the  average  difference  of  consecutive  logarithms  on 
the  same  line,  but  for  a  given  case  the  difference  needs  to  be 
verified  by  actual  subtraction,  at  least  so  far  as  the  last  figure 
is  concerned.  The  lower  part  of  the  page  contains  a  complete 
list  of  differences,  with  their  multiples  divided  by  10. 

To  find  the  logarithm  of  a  number  having  six 
figures :— Take  out  the  mantissa  for  the  four  superior  places 
directly  from  the  table,  and  find  the  difference  between  this 
mantissa  and  the  next  greater  in  the  table.  Add  to  the  man- 
tissa taken  out  the  quantity  found  in  the  table  of  proportional 
parts,  opposite  the  difference,  and  in  the  column  headed  by  the 
fifth  figure  of  the  number;  also  add  T^  the  quantity  in  the  col- 
umn headed  by  the  sixth  figure.  The  sum  is  the  mantissa 
required,  to  which  must  be  prefixed  a  decimal  point  and  the 
proper  characteristic. 


EXPLANATION"   OF   TABLES.  47 

Example.—  Find  the  log  of  23.4275. 

For  2342  mantissa  is  369587 

"    diff.  185  col.  7  129.5 

"       "       "     "    5  9.2 


Ans.  For  23.4275  log  is    1.369726 

The  decimals  of  the  corrections  are  added  together  to  deter- 
mine the  nearest  value  of  the  sixth  figure  of  the  mantissa. 

To  find  the  number  corresponding'  to  a  given 
logarithm. — If  the  given  mantissa  is  not  in  the  table  find  the 
one  next  less,  and  take  out  the  four  figures  corresponding  to  it; 
divide  the  difference  between  the  two  mantissas  by  the  tabu- 
lar difference  in  that  part  of  the  table,  and  annex  the  figures  of 
the  quotient  to  the  four  figures  already  taken  out.  Finally, 
place  the  decimal  point  according  to  the  rule  for  characteristics, 
prefixing  or  annexing  ciphers  if  necessary.  The  division  re- 
quired is  facilitated  by  the  table  of  proportional  parts,  \\  hich 
furnishes  by  inspection  the  figures  of  the  quotient. 

Example. — Find  the  number  of  which  the  logarithm  is 
8.263927  8.263927 

First  4  figures  1836  from  263873 

Diff.          540 
Tabular  diff.  =236          .  • .  5th  fig.  =2  47. 2 


6.80 
6th  fig.  =  3  7.08 

Ans.  No.  =  .0183623  or  183,623,000. 

The  number  derived  from  a  six-place  logarithm  is  not 
reliable  beyond  the  sixth  figure. 

At  the  end  of  table  XXIV.  is  a  small  table  of  logarithms  of 
numbers  from  1  to  100,  with  the  characteristic  prefixed,  for 
easy  reference  when  the  given  number  does  not  exceed  two 
digits.  But  the  same  mantissas  may  be  found  in  the  larger 
table. 

APPENDIX  X. — The  logarithmic  sine,  tangent, 
etc.  of  an  arc  is  the  logarithm  of  the  natural  sine,  tangent, 
etc.  of  the  same  arc,  but  with  10  added  to  the  characteristic  to 
avoid  negatives.  This  table  gives  log  sines,  tangents,  cosines, 
and  cotangents  for  every  minute  of  the  quadrant.  With  the 
number  of  degrees  at  the  left  side  of  the  page  are  to  be  road 
the  minutes  in  the  left-hand  column  ;  with  the  degrees  on 


48  STATISTICAL   METHODS. 

the  right-hand  side  are  to  be  read  the  minutes  in  the  right-hand 
column.  When  the  degrees  appear  at  the  top  of  the  page  the 
top  headings  must  be  observed,  when  at  the  bottom  those  at 
the  bottom.  Since  the  values  found  for  arcs  in  the  first  quad- 
rant are  duplicated  in  the  second,  the  degrees  are  given  from 
0°  to  180°.  The  differences  in  the  logarithms  due  to  a  change 
of  one  second  in  the  arc  are  given  in  adjoining  columns. 

To  find  the  log.  sin,  cos,  tan,  or  cot  of  a  given 
arc. :  Take  out  from  the  proper  column  of  the  table  the  log- 
arithm corresponding  to  the  given  number  of  degrees  and 
minutes.  If  there  be  any  seconds  multiply  them  by  the  ad- 
joining tabular  difference,  and  apply  their  product  as  a  cor- 
rection to  the  logarithm  already  taken  out.  The  correction  is 
to  be  added  if  the  logarithms  of  the  table  are  increasing  with 
the  angle,  or  subtracted  if  they  are  decreasing  as  the  angle  in- 
creases. In  the  first  quadrant  the  log  sines  and  tangents  in- 
crease, and  the  log.  cosines  and  cotangents  decrease  as  the 
angle  increases. 
Example.— Find  the  log  sin  of  9°  28'  20". 

Log  sin  of  9°  28'  is  9.216097 

Add  correction  20  X  12.62  252 

Ans.  9.216349 
Example.—  Find  the  log  cot  of  9°  28'  20". 

Log  cotan  of  9°  28'  is  10.777948 

Subtract  correction  20  X  12.97  259 

Ans.  10-777689 

To  find  the  angle  or  arc  corresponding  to  a 
given  logarithmic  sine,  tangent,  cosine,  or  co- 
tangent.— If  the  given  logarithm  is  found  in  the  proper 
column  take  out  the  degrees  and  minutes  directly;  if  not,  find 
the  two  consecutive  logarithms  between  which  the  given 
logarithm  would  fall,  and  adopt  that  one  which  corresponds  to 
the  least  number  of  minutes;  which  minutes  take  out  with  the 
degrees,  and  divide  the  difference  between  this  logarithm  and 
the  given  one  by  the  adjoining  tabular  difference  for  a  quo- 
tient, which  will  be  the  required  number  of  seconds. 

With  logarithms  to  six  places  of  decimals  the  quotient  is 
not  reliable  beyond  the  tenth  of  a  second. 


EXPLANATION    OF    TABLES.  49 

Example. — 9.383731  is  the  log  tan  of  what  angle? 
Next  less  9.383682  gives  13°  36' 

Diff.  49.00  -f-  9.20  =  05".3 


Ans.     13°  36'  05".3 

Example. — 9.249348  is  the  log  cos  of  what  angle? 
Next  greater  583  gives  79°  46' 

Diff.  235  -H  11.67  =  20M 


Ans.     79°  46'  20".l 

The  above  rules  do  not  apply  to  the  first  two  pages  of  this 
table  (except  for  the  column  headed  cosine  at  top)  because 
here  the  differences  vary  so  rapidly  that  interpolation  made  by 
them  in  the  usual  way  will  not  give  exact  results. 

On  tbe  first  two  pages,  the  first  column  contains  the  number 
of  seconds  for  every  minute  from  1'  to  2° ;  the  minutes  are 
given  in  the  second,  the  log.  sin.  in  the  third,  and  in  \kzfourth 
are  the  last  three  figures  of  a  logarithm  which  is  the  difference 
between  the  log  sin  and  the  logarithm  of  the  number  of  sec- 
onds in  the  first  column.  The  first  three  figures  and  the  char- 
acteristic of  this  logarithm  are  placed,  once  for  all,  at  the  head 
of  the  column. 

To  find  the  log  sin  of  an  arc  less  than  2°  given 
to  seconds. — Reduce  the  given  arc  to  seconds,  and  take  the 
logarithm  of  the  number  of  seconds  from  the  table  of  loga- 
rithms, and  add  to  this  the  logarithm  from  the  fourth  column 
opposite  the  same  number  of  seconds.  The  sum  is  the  log  sin 
required. 

The  logarithm  in  the  fourth  column  may  need  a  slight  inter- 
polation of  the  last  figure,  to  make  it  correspond  closely  to  the 
given  number  of  seconds. 

Example.— Find  the  log  sin  of  1°  39'  14".  4. 

1°  39'  14".4  =  5954".4  log  3.774838 

add  (q-l)  4.685515 

Ans.  log  sin  8.460353 

Log  tangents  of  small  arcs  are  found  in  the  same  way,  only 
taking  the  last  four  figures  of  (q  —  I)  from  the  fifth  column. 


50  STATISTICAL   METHODS. 

Example.—  Find  the  log  tan  of  0°  52'  35". 

add  (q  -  T)  4.685609 


52'  35"  =  (3120"  +  35")  =  3155"  log  3.498999 

'  '  t  —  I)   ' 


Ans.     log  tan  8. 184608 

To  find  the  log  cotangent  of  an  angle  less  than 
2°  given  to  seconds. — Take  from  the  column  headed  ( q-\-  Z) 
the  logarithm  corresponding  to  the  given  angle,  interpolating 
for  the  last  figure  if  necessary,  and  from  this  subtract  the  loga- 
rithm of  the  number  of  seconds  in  the  given  angle. 

Example.— Find  the  log  cotan  of  1°  44'  22".  5. 

g  +  I  15.314292 
6240"  +  22".  5  =  6262.5  log    3.796748 

Ans.     11.517544 

These  two  pages  may  be  used  in  the  same  way  when  the 
given  angle  lies  between  88°  and  92°,  or  between  178°  and  180°; 
but  if  the  number  of  degrees  be  found  at  the  bottom  of  the  page, 
the  title  of  each  column  will  be  found  there  also;  and  if  the 
number  of  degrees  be  found  on  the  right  hand  side  of  the  page, 
the  number  of  minutes  must  be  found  In  the  right  hand  col- 
umn, and  since  here  the  minutes  increase  upward,  the  number 
of  seconds  on  the  same  line  in  the  first  column  must  be  'dimin- 
ished by  the  odd  seconds  in  the  given  angle  to  obtain  the  num- 
ber whose  logarithm  Is  to  be  used  with  (q  ±  I)  taken  from  the 
table. 

Example.—  Find  the  log  cos  of  88°  41'  12". 5 

(q  -  I)  4.685537 
4740"  -  12*.5  =  4727.5  log  3.674631 

Ans.  8.360168 
Example.—  Find  the  log  tan  of  90°  30'  50". 

q+l  15.314413 

1800"  +  50"  ==  1850'  log    3.267172 

Ans.     12.047241 

To  find  the  arc  corresponding  to  a  given  log 
sin,  cos,  tan,  or  cotan  which  falls  within  the 
limits  of  the  first  two  pages  of  Table  X. 

Find  in  the  proper  column  two  consecutive  logarithms  be- 
tween which  the  given  logarithm  falls.  If  the  title  of  the 
given  function  is  found  at  the  top  of  that  column  read  the 


EXPLANATION   OF   TABLES.  51 

degrees  from  the  top  of  the  page ;  if  at  the  "bottom  read  from 
the  bottom. 

Find  the  value  of  (q  —  I)  or  (q  +  0,  as  the  case  may  require, 
corresponding  to  the  given  log  (interpolating  for  the  last  figure 
if  necessary).  Then  if  q  =  given  log  and  I  —  log  of  number  of 
seconds,  n,  in  the  required  arc,  we  have  at  once  I  =  q  —  (q  —  I) 
or  I  =  (q  +  0  —  <7>  whence  n  is  easily  found. 

Find  in  the  first  column  two  consecutive  quantities  between 
which  the  number  n  falls,  and  if  the  degrees  are  read  from 
the  left  hand  side  of  the  page,  adopt  the  less,  take  out  the 
minutes  from  the  second  column,  and  take  for  the  seconds 
the  difference  between  the  quantity  adopted  and  the  number 
n.  But  if  the  degrees  are  read  from  the  right  hand  side  of  the 
page,  adopt  the  greater  quantity,  take  out  the  minutes  on  the 
same  line  from  the  right-hand  column,  and  for  the  seconds 
take  the  difference  between  the  number  adopted  and  the  num- 
ber n. 

Eacample.— 11.734268  is  the  log  cot  of  what  arc? 
q  +  l  15.314376 

q  11.734268 

.-.     n  =  3802.8  "8.680108 

For  1°  adopt      3780.        giving  03' 

Difference  22".  8 

Ans.  1°  03'  22".8  or  178°  56'  37".2. 

Example.—  8.201795  is  the  log  cos  of  what  arc? 

q  -  I  4.685556 

q  8.201795 

.".       n-  3282".  8                                     3.516239 

For  89°  adopt  3300.       giving  05' 

Difference  17".  2 

Ans.  89°  05'  17*. 2  or  90°  54'  42*. 8. 


52  STATISTICAL   METHODS. 

I. -FORMULAS. 

M=^f)=v^_^  P.E.X=±  0.6745-^.  x=V-M. 

yn 

./2(a;2./)       / , —  <T 

n  **'  p'K<r=    4  v^r' 

^p  =     ^'^  -  0.7979<r.        P.^.  =  g  =  0.6745a. 


=   "2    -    *!*(+  ft)  = 


a  ~  Vl»  +  A)  = 


A^  (for  graduated  variates)  =      l 


A^  (for  integral  variates)  =  2^—^  .  100^,  where  fc  equals  the  number  of  classes. 

_  S(dev.  a;.dev.  y.f)  _ 
--  " 


Po  (spurious  correlation)  =  - 


+  V  1/tV  +  V 
7i  (index  of  heredity,  uniparental  inheritance)  =  p—  . 


x  =  p,—  7i2  +  Pa—^i    [biparental  inheritance:  unassortative  mating]. 
<ra  <r, 

=  P!  ~  Pl^3  .  -/ia  +  ^  ~  Pl1?  •  -  .  /i»    [biparental  inheritance;  assortative 
1  —  PJ-*       <ra  1  —  pi-*       (ra 

mating]* 


CERTAIN   CONSTANTS   AND   THEIR  LOGARITHMS.         53 
II.— CERTAIN  CONSTANTS  AND  THEIR  LOGARITHMS. 


Title. 

Symbol 

Number. 

Log. 

Ratio  of  circumference  to  diameter.  .  .  . 

IT 

1 

IT 

M^ 
i 

y;r 

V2^ 
1 
V2V 
e 

m 

1 
771 

3.1415927 
0.3183099 

1.7724538 
0.5641896 
2.506628 

0.3989422 
2.7182818 
0.4342945 

2.3025851 

0.4971499 
9.5028501 

0.2485749 
9.7514251 
0.399090 

9.6009100 
0.4342945 
9.6377843 

0.3622157 

Square  root  of  same  

Reciprocal  of  square  root  of  same  ...        . 

Square  root  of  2n-               

Modulus  of  common  system  of  logs  =  log  «... 
Reciprocal  of  same  —  hyp.  log  10  

Com.  log  x  —  m  X  hyp.  log  x,  or 
Com.log(com.logo;)=9.6377843+com.log(hyp.log:e) 
Hyp.  log  x  =  com.  log  x  X  —  ,    or 

Wl. 

Com.log(hyp.log#)==com.log(com.logo;)-f  0.3622157 
Circumference  of  circle  —    

2irr 
Trr* 
Wr 

»- 

najor  axis 
*  axis  of  el 

Area  of  circle                            »  ...        •  .  . 

Area  of  sector  (angle  of  arc  —  a°)  

minoi 

lipse. 

54 


STATISTICAL  METHODS. 


III.— TABLE  OF  ORDINATES  OF  NORMAL  CURVE,  OR  VALUES_OF 
—  CORRESPONDING  TO  VALUES  OF  -. 

yo  * 

x  =  deviation  from  mean,    y  —  frequency. 

o-  =  standard  deviation.       y0  =  — ^=.  =  maximum  frequency. 
a-  y2n 


X/V 

y/y<> 

x/* 

y/yQ 

*/* 

2//2/0 

x/a- 

y/y<> 

0 

1. 

0.8 

.7262 

1.6 

.2780 

2.8 

.0198 

0.1 

.9950 

0.9 

.6670 

1.7 

.2357 

3.0 

.0111 

0.2 

.9802 

1.0 

.6065 

1.8 

.1979 

3.2 

.0060 

0.3 

.9560 

1.1 

.5467 

1.9 

.1645 

3.4 

.0031 

0.4 

.9231 

1.2 

.4868 

2.0 

.1353 

3.6 

.0015 

0.5 

.8825 

1.3 

.4286 

2.2 

.0889 

3.8 

.0007 

0.6 

.8353 

1.4 

.3753 

2.4 

.0561 

4.0 

.0003 

0.7 

.7827 

1.5 

.3246 

2.6 

.0340 

5.0 

.000004 

YALUES   OF   THE   NORMAL   PROBABILITY    INTEGRAL.      55 

IV.— TABLE    OF  VALUES  OF    THE  NORMAL    PROBABILITY  INTEGRAL 

CORRESPONDING  TO  VALUES  OF  -  ;    OR  THE   FRACTION   OF  THE 

cr 

AREA    OF   THE    CURVE    BETWEEN  THE    LIMITS    0   AND  -f  -  OR  0 


Total  area  of  curve  assumed  to  be  10000. 
x  =  deviation  from  mean, 
cr  =  standard  deviation. 


X 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

A 

0.0 

0000 

0040 

0080 

0120 

0160 

0500 

0539 

0579 

0319 

0359 

40 

0.1 

0399 

0438 

0478 

0517 

0557 

0597 

0636 

0676 

0715 

0754 

40 

0.2 

0793 

0832 

0871 

0910 

0948 

0987 

1056 

1064 

1103 

1141 

39 

0.3 

1179 

1217 

1255 

1293 

1330 

1368 

1406 

1443 

1480 

1517 

38 

0.4 

1554 

1591 

1628 

1664 

1700 

1737 

1773 

1808 

1844 

1879 

36 

o.! 

1915 

1950 

1985 

2020 

2054 

2089 

2124 

2157 

2191 

2225 

34 

0.6 

2258 

2291 

2324 

2357 

2389 

2455 

2454 

2486 

2518 

2549 

32 

0.7 

2581 

2612 

2643 

2672 

2704 

2734 

5764 

2794 

2853 

2853 

30 

0.8 

2882 

2910 

2939 

2967 

2995 

3053 

3051 

3078 

3106 

3133 

28 

0.9 

3160 

3186 

3212 

3238 

3564 

3590 

3315 

3340 

3365 

3389 

& 

.0 

3414 

3438 

3461 

3485 

3509 

3532 

3555 

3577 

3600 

3622 

23 

.1 

3644 

3665 

3686 

3708 

3729 

3750 

3770 

3791 

3811 

3830 

21 

.2 

3850 

3869 

3888 

3906 

3955 

3944 

3962 

3980 

3997 

4015 

19 

.3 

4032 

4049 

4066 

4083 

4099 

4115 

4135 

4147 

4162 

4178 

17 

.4 

4193 

4208 

4222 

4237 

4551 

4565 

4279 

4292 

4306 

4319 

14 

1.5 

4332 

4345 

4358 

4370 

4383 

4395 

4406 

4418 

4429 

4441 

12 

1.6 

4452 

4463 

4474 

4485 

4496 

4506 

4516 

4526 

4536 

4545 

10 

1.7 

4554 

4564 

4573 

4582 

4591 

4600 

4608 

4617 

4655 

4633 

9 

1.8 

4641 

4648 

4656 

4664 

4671 

4678 

4686 

4693 

4700 

4706 

7 

1.9 

4713 

4720 

4756 

4732 

4738 

4744 

4750 

4756 

4762 

4767 

6 

2.0 

4773 

4778 

4783 

4788 

4794 

4799 

4804 

4808 

4813 

4817 

5 

2.1 

4822 

4826 

4830 

4834 

4838 

4842 

4846 

4850 

4854 

4858 

4 

2.2 

4861 

4865 

4868 

4872 

4875 

4878 

4881 

4884 

4887 

4890 

3 

2.3 

4893 

4896 

4899 

4901 

4904 

4906 

4909 

4911 

4914 

4916 

3 

2.4 

4918 

4921 

49:23 

4955 

4927 

4929 

4931 

4933 

4935 

4936 

2 

2.5 

4938 

4940 

4942 

4943 

4945 

4946 

4947 

4949 

4951 

4952 

2 

2.6 

4953 

4955 

4956 

4958 

4959 

4960 

4961 

4962 

4964 

4965 

1 

2.7 

4966 

4967 

4968 

4969 

4970 

4970 

4971 

4972 

4973 

4974 

1 

2.8 

4975 

4975 

4976 

4977 

4978 

4978 

4979 

4980 

4981 

4981 

0.5 

2.9 

4982 

4982 

4983 

4983 

4984 

4984 

4985 

4985 

4986 

4986 

0.5 

3 

4987 

4991 

4993 

4995 

4997 

4998 

4999 

4999 

4999 

5000 

GO 

5000 

56 


STATISTICAL   METHODS. 
V.— TABLE  OF  LOG    T  FUNCTIONS  OF  p. 


p 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1.00 

9750 

9500 

q 
«251 

9003 

8755 

8509 

8263 

8017 

7773 

1.01 

9.997529 

7285 

7043 

b801 

6560 

6320 

6080 

5841 

5602 

5365 

1.02 

5128 

4892 

4656 

4421 

4187 

3953 

3721 

3489 

3257 

3026 

1.03 

2796 

2567 

2338 

2110 

1883 

1656 

1430 

1205 

0981 

0775 

1.04 

0533 

0311 

0089 

9868 

§647 

§427 

§208 

§989 

8-72 

§554 

1.05 

9.988338 

8122 

7907 

7692 

7478 

.  7265 

7052 

6841 

6629 

6419 

1.06 

6209 

6000 

5791 

5583 

5378 

5169 

4963 

4758 

4553 

4349 

1.07 

4145 

3943 

3741 

3539 

3338 

3138 

2939 

2740 

2541 

2344 

1.08 

2147 

1951 

1755 

1560 

1365 

1172 

0978 

0786 

0594 

0403 

1.09 

0212 

0022 

§833 

§644 

§456 

§269 

§082 

§900 

§710 

§525 

1.10 

9.978341 

8157 

7974 

7791 

7610 

7428 

7248 

7068 

6888 

6709 

1.11 

6531 

6354 

6177 

6000 

5825 

5650 

5475 

5301 

5128 

4955 

I.I* 

4783 

4612 

4441 

4271 

4101 

3932 

3764 

3596 

3429 

3262 

1.13 

3096 

2931 

2766 

2602 

2438 

2275 

2113 

1951 

1790 

1629 

1.14 

1469 

1309 

1150 

0992 

0835 

0677 

0521 

0365 

0210 

0055 

1.15 

9.969901 

9747 

9594 

9442 

9290 

9139 

8988 

8838 

8688 

8539 

1.16 

8390 

8243 

8096 

7949 

7803 

7658 

7513 

7369 

7225 

7082 

1.17 

6939 

6797 

6655 

6514 

6374 

6234 

6095 

5957 

5818 

5681 

1.18 

5544 

5408 

5272 

5137 

5002 

4868 

4734 

4601 

4469 

4337 

1.19 

4205 

4075 

3944 

3815 

3686 

3557 

3429 

3302 

3175 

3048 

1.20 

2922 

2797 

2672 

2548 

2425 

2302 

2179 

2057 

1936 

1815 

KM 

1695 

1575 

1456 

1337 

1219 

1101 

0984 

0867 

0751 

0636 

1.22 

0521 

0407 

0293 

0180 

0067 

9955 

8843 

9732 

9621 

9511 

1.23 

9.959401 

9292 

9184 

9076 

8968 

8861 

8755 

8649 

8544 

8439 

1.24 

8335 

8231 

8128 

8025 

7923 

7821 

7720 

7620 

7520 

7420 

1.25 

7321 

7223 

7125 

7027 

6930 

6834 

6738 

6642 

6547 

6453 

1.26 

6359 

6267 

6173 

6081 

5989 

5898 

5807 

5716 

5627 

5537 

1.27 

5449 

5360 

5273 

5185 

5099 

5013 

4927 

4842 

4757 

4673 

1,28 

4589 

4506 

4423 

4341 

4-259 

4178 

4097 

4017 

3938 

3858 

1.29 

3780 

3702 

3624 

3547 

3470 

3394 

3318 

3243 

3168 

3094 

1.30 

3020 

2947 

2874 

2802 

2730 

2659 

2588 

2518 

2448 

2379 

1.31 

2310 

2242 

2174 

2106 

2040 

1973 

1907 

1842 

1777 

1712 

1.32 

1648 

1585 

1522 

1459 

1397 

1336 

1275 

1214 

1154 

1094 

1.33 

1035 

0977 

0918 

0861 

0803 

0747 

0690 

0634 

0579 

0524 

1.34 

0470 

0416 

0362 

0309 

0257 

0205 

0153 

0102 

0051 

0001 

1.35 

9.  949951 

9902 

9853 

9805 

9757 

9710 

9663 

9617 

9571 

9525 

1.36 

9430 

9435 

9391 

9348 

9304 

9262 

9219 

9178 

9136 

9095 

1.37 

9054 

9015 

8975 

8936 

8898 

8859 

8822 

8785 

8748 

8711 

1.38 

8676 

8640 

8605 

8571 

8537 

8503 

8470 

8437 

8405 

8373 

1.39 

8342 

8311 

8280 

8250 

8221 

8192 

8163 

8135 

8107 

8080 

1.40 

8053 

8026 

8000 

7975 

7950 

7925 

7901 

7877 

7854 

7831 

1.41 

7808 

7786 

7765 

7744 

7723 

7703 

7683 

7664 

7645 

7626 

1.42 

7608 

7590 

7573 

7556 

7540 

7524 

7509 

7494 

7479 

7465 

1.43 

7451 

7438 

7425 

7413 

7401 

7389 

7378 

7368 

7358 

7348 

1.44 

7338 

7329 

7321 

7312 

7305 

7298 

7291 

7284 

7278 

7273 

1.45 

7268 

7263 

7259 

7255 

7251 

7248 

7246 

7244 

7242 

7241 

1.46 

7240 

7239 

7239 

7240 

7241 

7242 

7243 

7245 

7248 

7251 

1.47 

7254 

7258 

7262 

7266 

7271 

7277 

7282 

7289 

7295 

7302 

1.48 

7310 

7317 

7326 

7334 

7343 

7353 

7363 

7373 

7384 

7395 

1.49 

7407 

7419 

7431 

7444 

7457 

7471 

7485 

7499 

7515 

7529 

TABLE   OF  LOG  P  FUKCTIOI^S. 
V.— TABLE  OF  LOG  r  FUNCTIONS  OF  p. 


P 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1.50 

9.947545 

7561 

7577 

7594 

7612 

7629 

7647 

7666 

76S5 

7704 

1.51 

7724 

7744 

7764 

7785 

7806 

7828 

7850 

7873 

7896 

7919 

1.52 

7943 

7967 

7991 

8016 

8041 

8067 

8093 

8120 

8146 

8174 

1.53 

8201 

8229 

8258 

8287 

8316 

8346 

8376 

8406 

8437 

8468 

1.54 

8500 

8532 

8564 

8597 

8630 

8664 

8698 

8732 

8767 

8802 

.55 

8837 

8873 

8910 

8946 

8983 

9021 

9059 

9097 

9135 

9174 

.56 

9214 

9254 

9294 

9334 

9375 

9417 

9458 

9500 

9543 

9580 

.57 

98  .'9 

9672 

9716 

9761 

9806 

9851 

9896 

9942 

9989' 

5035 

.58 

9.950082 

0130 

0177 

0225 

0274 

0323 

0372 

0422 

0472 

052-2 

.59 

0573 

0624 

0676 

0728 

0780 

0833 

0886 

0939 

0993 

1047 

.60 

1102 

1157 

1212 

1268 

1324 

1380 

1437 

1494 

1552 

1610 

.61 

1668 

1727 

1786 

1845 

1905 

1965 

2025 

2086 

2147 

2209 

.62 

2271 

2333 

2396 

2459 

2522 

2586 

2650 

2715 

2780 

2845 

.63 

2911 

2977 

3043 

3110 

3177 

3244 

3312 

3380 

3449 

3517 

.64 

3587 

3656 

3726 

3797 

3867 

3938 

4010 

4081 

4154 

4226 

.65 

4299 

4372 

4446 

4519 

4594 

4668 

4743 

4819 

4894 

4970 

.66 

5047 

5124 

5201 

5278 

5356 

5434 

5513 

5592 

5671 

5740 

.67 

5830 

5911 

5991 

6072 

6154 

6235 

6317 

6400 

6482 

6566 

.68 

6649 

6733 

6817 

6901 

6986 

7072 

7157 

7243 

7322 

7416 

.69 

7503 

7590 

7678 

7766 

7854 

7943 

803-2 

8122 

8211 

8301 

.70 

8391 

8482 

8573 

8664 

8756 

8848 

8941 

9034 

9127 

92-20 

.71 

9314 

9409 

9502 

9598 

9ti93 

9788 

9884 

9980 

5077 

5174 

.72 

9.960271 

0369 

0467 

0565 

0664 

0763 

0862 

0961 

1061 

1162 

.73 

1262 

1363 

1464 

1566 

1668 

1770 

1873 

1976 

2079 

2183 

.74 

2287 

2391 

2496 

2601 

2706 

2812 

2918 

3024 

3131 

3238 

.75 

3345 

3453 

3561 

3669 

3778 

3887 

3996 

4105 

4215 

43-26 

.76 

4436 

4547 

4659 

4770 

4882 

4994 

5107 

5220 

5333 

5447 

.77 

5561 

5675 

5789 

5904 

6019 

6135 

6251 

6367 

6484 

6600 

.78 

6718 

6835 

6953 

7071 

7189 

7308 

7427 

7547 

76C6 

7787 

.79 

7907 

8023 

8149 

8270 

8392 

8514 

8636 

8759 

8882 

9005 

.80 

9129 

9253 

9377 

9501 

9626 

9751 

9877 

5008 

5129 

5255 

.81 

9.970383 

0509 

0637 

0765 

0893 

1021 

1150 

1279 

1408 

1538 

.82 

1668 

1798 

1929 

2060 

2191 

2322 

2454 

2586 

2719 

285-2 

.83 

2985 

3118 

3252 

3386 

3520 

3655 

3790 

3925 

4061 

4197 

.84 

4333 

4470 

4606 

4744 

4881 

5019 

5157 

5295 

5434 

5573 

.85 

5712 

5852 

5992 

6132 

6273 

6414 

6555 

6697 

6838 

6980 

.86 

7123 

7266 

7408 

7552 

7696 

7840 

7984 

8128 

8273 

8419 

.87 

8564 

8710 

8856 

900-2 

9149 

9296 

9443 

9591 

9739 

9887 

.88 

9.980036 

0184 

0333 

0483 

0633 

0783 

0933 

1084 

1234 

1386 

.89 

1537 

1689 

1841 

1994 

2147 

2299 

2453 

2607 

2761 

2915 

.90 

3069 

3224 

3379 

3535 

3690 

3846 

4003 

4159 

4316 

4474 

.91 

4631 

4789 

4947 

5105 

5264 

5423 

5582 

5742 

5902 

606-2 

.92 

6223 

6383 

6544 

6706 

6867 

7029 

7192 

7354 

7517 

7680 

.93 

7844 

8007 

8171 

8336 

8500 

8665 

8830 

8996 

9161 

9327 

.94 

9494 

9660 

9827 

9995 

5162 

5330 

5498 

5666 

5835 

1004 

.95 

9.991173 

1343 

1512 

1683 

1853 

2024 

2195 

2366 

2537 

2709 

.96 

2881 

3054 

32-27 

3399 

3573 

3746 

3920 

4094 

4269 

4443 

.97 

4618 

4794 

4969 

5145 

53-21 

5498 

5674 

5851 

6029 

6206 

.98 

6384 

6562 

6740 

6919 

7078 

7277 

7457 

7637 

7817 

7997 

.99 

8178 

8359 

8540 

8722 

8903 

9085 

9268 

9450 

9633 

9816 

58 


STATISTICAL    METHODS. 


VI.-TABLE  OF  REDUCTION  FROM  COMMON  TO  METRIC  SYSTEM. 


Inches  to  Millimeters. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

25.40 

50. 

80 

76.20 

101.60 

127.00 

152.  4C 

177 

.80 

2( 

)3.20 

228.60 

10 

279.40 

304. 

80 

330.19 

355.59 

380.99 

406.  3£ 

431 

.79 

4; 

>7  19 

482.59 

20 

533.39 

558. 

79 

584.19 

609.59 

634.99 

660.32 

685 

.79 

711.19 

736.59 

30 

787.39 

812. 

79 

838  19 

863.59 

888.99 

914.32 

939 

.78 

9( 

55.18 

990.58 

40 

1041.4 

1066. 

8 

1092.2 

1117.6 

1143.0 

1168.4 

1193 

.8 

1219.2 

1244.6 

50 

1295.4 

1320. 

8 

1346.2 

1371.6 

1397.0 

1422.4 

1447 

.8 

14 

~3  2 

1498.6 

liO 

1549.4 

1574. 

8 

1600.2 

16-25.6 

1651.0 

1676.4 

1701 

.8 

17: 

17.2 

1752.6 

70 

1803.4 

1828. 

H 

1854.2 

1879.6 

1905.0 

1930.4 

1955 

.8 

19* 

U.2 

2006.6 

80 

2057.4 

2082. 

8 

2108.2 

2133.6 

2159.0 

2184.4 

2209 

.8 

2235.2 

2260.6 

90 

2311.4 

2336. 

8 

2362.2 

2387.6 

2413.0 

2438.4 

2463 

.8 

2489.2 

2514.6 

Twelfths. 

Sixteenths. 

1/12 

2.12 

7/12 

14.82 

1/16 

1.59 

5/16 

7.94 

9/16 

14.29 

13/16    20.64 

2/12 

4.23 

8/12 

16.93 

1/8 

3.17 

3/8 

9.52 

5/8 

15.87 

7/8     22.22 

3/12 

6.35 

9/12 

19.05 

3/16 

4.76 

7/16 

11.11 

11/16 

17.46 

15/16    23.81 

4/12 

8.47 

10/12 

21.17 

1/4 

6.35 

1/2 

12.70 

3/4 

19.0J 

1        25.  41 

5/12 

10.58 

11/12 

23.28 

6/12 

12.70 

12/12 

25.40 

FIRST   TO   SIXTH    POWERS   OF    INTEGERS. 


5.9 


TABLE  VIL— FIRST  TO   SIXTH  POWERS  OF  INTEGERS  FROM  1  TO  30. 


Powers. 

First. 

Second. 

Third. 

Fourth. 

Fifth. 

Sixth. 

1 

.  1 

1 

1 

1 

1 

2 

4 

8 

16 

32 

64 

8 

9 

27 

81 

243 

729 

4 

16 

64 

256 

1024 

4096 

5 

25 

125 

625 

3125 

15625 

6 

36 

216 

1296 

7776 

46656 

7 

49 

343 

2401 

16807 

117649 

8 

64 

512 

4096 

32768 

262144 

9 

81 

729 

6561 

59049 

531441 

10 

100 

1000 

10000 

100000 

1000000 

11 

121 

1331 

14641 

161051 

1771561 

12 

144 

1728 

20736 

248832 

2985984 

13 

169 

2197 

28561 

371293 

4826809 

14 

196 

2744 

38416 

537824 

7529536 

15 

225 

3375 

50625 

759375 

11390625 

16 

256 

4096 

65536 

1048576 

1G777216 

17 

289 

4913 

83521 

1419857 

24137569 

18 

324 

5832 

104976 

1889568 

34012224 

19 

361 

6859 

1303-,>1 

2476099 

47045881 

20 

400 

8000 

160000 

3200000 

64000000 

21 

441 

9261 

194481 

4084101 

85766124 

22 

484 

10648 

234256 

5153632 

113379904 

23 

529 

12167 

279841 

6436343 

148035889 

24 

576 

13824 

331776 

7962624 

191102976 

25 

625 

156-J5 

390625 

9765625 

244140625 

26 

676 

17576 

456976 

11881376 

308915776 

27 

7*9 

19683 

531441 

14348907 

387420489 

28 

784 

21952 

614656 

17210368 

481890304 

29 

841 

24389 

707281 

30511149 

594823321 

30 

900 

27000 

810000 

24300000 

729000000 

TABLE   VIII. — SQUARES,    CUBES,   SQUARE   ROOTS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

1 

1 

1 

1.0000000 

1.0000000 

1.000000000 

2 

4 

8 

1.4142136 

1.2599210 

.500000000 

3 

9 

27 

1.7320508 

1.4422496 

.333333333 

4 

16 

64 

2.0000000 

1.5874011 

.250000000 

5 

25 

125 

2.2360680 

1.7099759 

.200000000 

6 

36 

216 

2  4494897 

1.8171206 

.166666667 

7 

49 

343 

2.6457513 

1.9129312 

.142857143  - 

8 

64 

512 

2.8284271 

2.0000000 

.125000000 

9 

81 

729 

3.0000000 

2.0800837 

.111111111 

10 

100 

1000 

3.1622777 

2.1544347 

.100000000 

11 

121 

1331 

3.3166248 

2.2239801 

.090909091 

12 

144 

1728 

3.4641016 

2.2894286 

.083333333 

13 

169 

2197 

3.6055513 

2.3513347 

.076923077 

14 

196 

2744 

3.7416574 

2.4101422 

.071428571 

15 

225 

3375 

3.8729833 

2.4662121 

.066666667 

16 

256 

4096 

4.0000000 

2.5198421 

.062500000 

17 

289 

4913 

4.1231056 

2.5712816 

.058823529 

18 

324 

5832 

4.2426407 

2.6207414 

.055555556 

19 

361 

6859 

4.3588989 

2.6684016 

.052631579 

20 

400 

8000 

4.4721360 

2.7144177 

.050000000 

21 

441 

9261 

4.5825757 

2.7589243 

.047619048 

22 

484 

10648 

4.6904158 

2.8020393 

.045454545 

23 

529 

12167 

4.7958315 

2.8438670 

.043478261 

24 

576 

13824 

4.8989795 

2.8844991 

.041666667 

25 

625 

15625 

5.0000000 

2.9240177 

.040000000 

26 

676 

17576 

5.0990195 

2.9624960 

.038461538 

27 

729 

19683 

5.1961524 

3.0000000 

.037037037 

28 

784 

21952 

5.2915026 

3.0365889 

.035714286 

29 

841 

24389 

5.3851648 

3.0723168 

.034482759 

30 

900 

27000 

5.4772256 

3.1072325 

.033333333 

31 

961 

29791 

5.5677644 

3.1413806 

.032258065 

32 

1024 

32768 

5.6568542 

3.1748021 

.031250000 

33 

1089 

35937 

5.7445626 

3.2075343 

030303030 

34 

1156 

39304 

5.8309519 

3.2396118 

.029411765 

35 

1225 

42875 

5.9160798 

3.2710663 

.028571429 

36 

1296 

46656 

6.0000000 

3.3019272 

.027777778 

37 

1369 

50653 

6.0827625 

3.3322218 

.027027027 

38 

1444 

54872 

6.1644140 

3.3619754 

.026315789 

39 

1521 

59319 

6.2449980 

3.3912114 

.025641026 

40 

1600 

64000 

6.3245553 

3.4199519 

.025000000 

41 

1681 

68921 

6.4031242 

3.4482172 

.024390244 

42 

1764 

74088 

6.4807407 

3.4760266 

.023809524 

43 

1849 

79507 

6.5574385 

3.50&3981 

.023255814 

44 

1936 

85184 

6.6332496 

3.5303483 

.022727273 

45 

2025 

91125 

6.7082039 

3.5568933 

.022222222 

46 

2116 

97336 

6.7823300 

3.5830479 

.021739130 

47 

2209 

103823 

6.8556546 

3.6088261 

.021276600 

48 

2304 

110592 

6.9282032 

3.6342411 

.020833333 

49 

2401 

117649 

7.0000000 

3.6593057 

.020408163 

50 

2500 

125000 

7.0710678 

3.6840314 

.020000000 

51 

2601 

132651 

7.1414284 

3.7084298 

.019607843 

52 

2704 

140608 

7.2111026 

3.7325111 

.019230769 

53 

2809 

148877 

7.2801099 

3.7562858 

.018867925 

54 

2916 

157464 

7.3484692 

3.7797631 

.018518519 

55 

3025 

166375 

7.4161985 

3.8029525 

.018181818 

56 

3136 

175616 

7.4833148 

3.8258624 

.017857143 

57 

3249 

185193 

7.5498344 

3.8485011 

.017543860 

58 

3364 

195112 

7.6157731 

3.8708766 

.017241379 

59 

3481 

205379 

7.6811457 

3.8929965 

.016949153 

60 

3600 

216000 

7.7459667 

3.9148676 

.016666667 

61 

3721 

226981 

7.8102497 

3.9364972 

.016393443 

62 

3844 

238328 

7.8740079 

3.9578915 

.016129032 

60 


CUBE  ROOTS,  AND  RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

63 

3969 

250047 

7.9372539 

3.9790571 

.015873016 

64 

4096 

262144 

8.0000000 

4.0000000 

.015625000 

65 

4225 

274625 

8.0622577 

4.0207256 

.015384615 

66 

4356 

287496 

8.1240384 

4.0412401 

.015151515 

67 

4489 

300763 

8.1853528 

4.0015480 

.014925373 

68 

4624 

314432 

8.2462113 

4.0816551 

.014705882 

69 

4761 

328509 

8.3066239 

4.1015661 

.014492754 

70 

4900 

343000 

8.3666003 

4.1212853 

.014285714 

71 

5041 

357911 

8.4261498 

4.1408178 

.014084507 

72 

5184 

373248 

8.4852814 

4.1601676 

.013888889 

73 

5329 

38901  r 

8.5440037 

4.1793390 

013698630 

74 

5476 

405224 

8.6023253 

4.1983364 

.013513514 

75 

5625 

421875 

8.6602540 

4.2171633 

.013333333 

76 

5776 

438976 

8.7177979 

4.2358236 

.013157895 

5929 

456533 

8.7749644 

4.2543210 

.012987013 

78 

6084 

474552 

8.8317609 

4.2726586 

.012820513 

79 

6241 

493039 

8.8881944 

4.2908404 

.012658228 

80 

6400 

512000 

8.9442719 

4.3088695 

.012500000 

81 

6561 

531441 

9.0000000 

4.3267487 

.012345679 

$2 

6724 

551308 

9.0553851 

4.3444815 

.012195122 

83 

6889 

571787 

9.1104336 

4.3620707 

.012048193 

84 

7056 

592704 

9.1651514 

4.3795191 

.011904762 

85 

7225 

614125 

9.2195445 

4.3968296 

.011764706 

86 

7396 

636056 

9.2736185 

4.4140049 

.011627907 

87 

7569 

658503 

9.3273791 

4.4310476 

.011494253 

88 

7744 

681472 

9.3808315 

4.4479602 

.011363636 

M 

7921 

704969 

9.4339811 

4.4647451 

.011235955 

90 

8100 

729000 

9.4868330 

4.4814047 

.011111111 

91 

8281 

753571 

9.5393920 

4.4979414 

.010989011 

92 

8464 

778688 

9.5916630 

4.5143574 

.010869565 

93 

8649 

804357 

9.6436508 

4.5306549 

.010752688 

94 

8836 

830584 

9.6953597 

4.5468359 

.010638298 

95 

9025 

857375 

9.7467943 

4.5629026 

.010526316 

96 

9216 

884736 

9.7979590 

4.5788570 

.010416667 

97 

9409 

912673 

9.8488578 

4.5947009 

.010309278 

98 

9604 

941192 

9.8994949 

4.6104363 

.010204082 

99 

9801 

970299 

9.9498744 

4.6260650 

.010101010 

100 

10000 

1000000 

10.0000000 

4.6415888 

.010000000 

101 

10201 

1030301 

10.0498756 

4.6570095 

.009900990 

102 

10404 

1001208 

10.0995049 

4.6723287 

.009803922 

103 

10609 

1092727 

10.1488916 

4.6875482 

.009708738 

104 

10816 

1124864 

10.1980390 

4.7026694 

.009615385 

105 

11025 

1157625 

10.2469508 

4.7176940 

.009523810 

106 

11236 

1191016 

10.2956301 

4.7326235 

.009433962 

107 

11449 

1225043 

10.3440804 

4.7474594 

.009345794 

108 

11664 

1259712 

10.3923048 

4.7622032 

.00/259259 

109 

11881 

1295029 

10.4403065 

4.7768562 

.009174312 

110 

12100 

1331000 

10.4880885 

4.7914199 

.009090909 

111 

12321 

1307631 

10.5356538 

4.8058955 

.009009009 

112 

12544 

1404928 

10.5830052 

4.8202845 

.  0089285  •! 

113 

12709 

1442897 

10.6301458 

4.8345881 

.008849558 

114 

12990 

1481544 

10.C770783 

4.8488076 

.008771930 

115 

13225 

1520875 

10.7238053 

4.8629442 

.008695652 

116 

13456 

1560896 

10.7703296 

4.8769990 

.008620690 

117 

13689 

1601613 

10.81665:58 

4.8909732 

.008547009 

118 

13924 

1643032 

10.8627805 

4.9048681 

.008474576 

119 

14161 

1685159 

10.9087121 

4.9186847 

.008403361 

120 

14400 

1728000 

10.9544512 

4.9324242 

.008333333 

121 

14641 

1771561 

11.  oo;  loooo 

4.9460874 

.008264463 

1S2 

14884 

1815848 

11.0453610 

4.9596757 

.008196721 

123 

15129     1860867 

11.0905365 

4.9731898 

.008130081 

124 

15376     1906624 

11.1355287 

4.9866310 

.008064516 

61 


TABLE   VIII. — SQUARES,    CUBES,    SQUARE    ROOTS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

125 

15625 

1953125 

11.1803399 

5.0000000 

.008000000 

126 

15876 

2000376 

11.2249722 

5.0132979 

-007936508 

127 

16129 

2048383 

11.2694277 

5.0265257 

.007874016 

128 

16384 

2097152 

11.3137085 

5.0390842 

.007812500 

129 

16641 

2146689 

11.3578167 

5.0527743 

.007751938 

130 

16900 

2197000 

11.4017543 

5.0657970 

.007692308 

131 

17161 

2248091 

11.4455231 

5.0787531 

.007633588 

132 

17424 

2299968 

11.4891253 

5.0916434 

.007575758 

133 

17689 

2352637 

11.5325626 

5.1044687 

.007518797 

134 

17956 

2406104 

11.5758369 

5.1172299 

.007462687 

135 

18225 

2460375 

11.6189500 

5.1299278 

.007407407 

136 

18496 

2515456 

11.6619038 

5.1425632 

.007352941 

137 

18769 

2571353 

11.7046999 

5.1551367 

.007299270 

138 

19044 

2628072 

11.7473401 

5.1676493 

.007246377 

139 

19321 

2685619 

11.7898261 

5.1801015 

.007194245 

140 

19600 

2744000 

11.8321596 

5.1924941 

.007142857 

141 

19881 

2803221 

11.874:3421 

5.2048279 

.007092199 

142 

20164 

2863288 

11.9163753 

5.2171034 

.007042254 

143 

20449 

2924207 

11.9582607 

5.2293215 

.006993007 

144 

20736 

2985984 

12.0000000 

5.2414828 

.006944444 

145 

21025 

3048625 

12.0415946 

5.2535879 

.006896552 

146  |   21316 

3112136 

12.0830460 

5.2656374 

.006849315 

147 

21609 

3176523 

12.1243557 

5.2776321 

.006802721 

148 

21904 

3241792 

12.1655251 

5.2895725 

.006756757 

149 

22201 

3307949 

12.2065556 

5.3014592 

.006711409 

150 

22500 

3375000 

12.2474487 

5.3132928 

.006666667 

151 

22801 

3442951 

12.2882057 

5.32507*40 

.006622517 

152 

23104 

3511808 

12.3288280 

5.3368033 

.006578947 

153 

23409 

3581577 

12.3693169 

5.3484812 

.006535948 

154 

23716 

3652264 

12.4096736 

5.3601084 

.006493506 

155 

24025 

3723875 

12.4498996 

5.3716854 

.006451613 

156 

24336 

3796416 

12.4899960 

5.3832126 

.006410256 

157 

24649 

3869893 

12.5299641 

5  3946907 

.006369427 

158 

24964 

3944312 

12.5698051 

5.4061202 

.006329114 

159 

25281 

4019679 

12..  0095203 

5.4175015 

.006289308 

160 

25600 

4096000 

12.6491106 

5.4288352 

.006250000 

161 

25921 

4173281 

12.6885775 

5.4401218 

.006211180 

162 

26244 

4251528 

12.7279221 

5.4513618 

.006172840 

163 

26569 

4330747 

12.7671453 

5.4625556 

.006134969 

164 

26896 

4410944 

12.8062485 

5.4737037 

.006097561 

165 

27225 

4492125 

12.8452326 

5.4848066 

.006060606 

166 

27556 

4574296 

12.8840987 

5.4958647 

.006024096 

167 

27889 

4657463 

12.9228480 

5.5068784 

.005988024 

168 

28224 

4741632 

12.9614814 

5.5178484 

.005952381 

169 

28561 

4826809 

13.0000000 

5.5287748 

.005917160 

170 

28900 

4913000 

13.03S4048 

5.5396583 

.005882353 

171 

29241 

5000211 

13.0766968 

5.5504991 

.005847953 

172 

29584 

5088448 

13.1148770 

5.5612978 

.005813953 

173 

29929 

5177717 

13.1529464 

5.5720546 

.005780347 

174 

30276 

5268024 

13.1909060 

5.5827702 

.005747126 

175 

30625 

5359375 

13.2287566 

5  .  5934447 

.005714286 

176 

30976 

5451776 

13.2664992 

5.6040787 

.005681818 

177 

31329 

5545233 

13.3041347 

5.6146724 

.005649718 

178 

31684 

5639752 

13.3416641 

6.6252263 

.005617978 

179 

32041 

5735339 

13.3790882 

5.6357408 

.005586592 

180 

32400 

5832000 

13.4164079 

5.6462162 

.005555556 

181 

32761 

5929741 

13.453(5240 

5.6566528 

.005524862 

182 

33124 

6028568 

13.4907376 

5.6670511 

.005494505 

183 

33489 

6128487 

13.5277493 

5.6774114 

.005464481 

184 

33856 

6229504 

13.5646600 

5.6877340 

.005434783 

185 

34225 

6331625 

13.6014705 

5.6980192 

.005405405 

186 

34596 

6434856 

13.6381817 

5.7082675 

.005376344 

CUBE  ROOTS,  AND  RECIPROCALS. 


K, 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

187 

34969 

6539203 

13.6747943 

5.7184791 

.005347594 

188 

35344 

6644672 

13.7113092 

5.7286543 

.005319149 

189 

35721 

6751269 

13.7477271 

5.7387936 

.005291005 

190 

36100 

6859000 

13.7840488 

5.7488971 

.005263158 

191 

36481 

6967871 

13.8202750 

5.7589652 

.005235602 

192 

36864 

7077888 

13.8564065 

5.7689982 

.005208333 

193 

37249 

7189057 

13.8924440 

5.7789966 

.005181347 

194 

37036 

7301384 

13.9283883 

5.7889604 

.005154639 

195 

38025 

7414875 

13.9642400 

5.7988900 

.005128205 

196 

38416 

7529536 

14.0000000 

5.8087857 

.005102041 

197 

38809 

7645373 

14.  0356688 

5.8186479 

.005076142 

198 

39204 

7762392 

14.0712473 

5.8284767 

.005050505 

199 

39601 

7880599 

14.1067360 

5.8382?25 

.005025126 

200 

40000 

8000000 

14.1421356 

5.8480355 

.005000000 

201 

40401 

8120601 

14.1774469     5.8577660 

.004975124 

202 

40804 

8242408 

14.2126704 

5.8674643 

.004950495 

203 

41209 

8365427 

14.247S068 

5.8771307 

.004926108 

204 

41616 

8489664 

14.2828569 

5.8867653 

.004901961 

205 

42025 

8615125 

14.3178211 

5.8963685 

.004878049 

206 

42436 

8741816 

14.3527001 

5.9059406 

.004854369 

207 

42849 

8869743 

14.3874946 

5.9154817 

.004830918 

208 

43264 

8998912 

14.4222051 

5.9249921 

.004807692 

209 

43681 

9129329 

14.4568323 

5.9344721 

.004784689 

210 

44100 

9261000 

14.4913767 

5.9439220 

.004761905 

211 

44521 

9393931 

14.5858890 

5.9533418 

.004739336 

212 

44944 

9528128 

14.5602198 

5.9627320 

.004716981 

213 

45369 

9663597 

I4.8W5195 

5.9720926 

.004694836 

214 

45796 

9800344 

14.6287388 

5.9814240 

.004672897 

215 

46225 

9938375 

14.6628783 

5.9907264 

.004651163 

216 

46656 

10077696 

14.6969385 

6.0000000 

.004629630 

217 

47089 

10218313 

14.7309199 

6.0092450 

.004608295 

218 

47524 

10360232 

14.7648231 

6.0184617 

.004587156 

219 

47961 

10503459 

14.7986486 

6.0276502 

.004566210 

220 

48400 

10648000 

14.8323970 

6.0368107 

.004545455 

221 

48841 

10793861 

14.8660687  ;   6.0459435 

.004524887 

222 

49284 

10941048 

14.8996644 

6.0550489 

.004504505 

223 

49729 

11089567 

14.9331845 

6.0641270 

.004484306 

224 

50176 

11239424 

14.9666295 

6.0731779 

.004464286 

225 

50625 

11390625 

15.0000000 

6.0822020 

.004444444 

226 

51076 

11543176 

15.0332964 

6.0911994 

.004424779 

227 

51529 

11697083 

15.0665192 

6.1001702 

.004405286 

228 

51984 

11852352 

15.0996689 

6.1091147 

.004385965 

229 

52441 

12008989 

15.1327460 

6.1180332 

.004366812 

230 

52900 

12167000 

15.1657509 

6.1269257 

.004347826 

231 

53361 

12326391 

15.1986842 

6.1357924 

.004329004 

232 

53824 

12487168 

15.2315462 

6.1446337 

.004310345 

233 

54289 

12649337 

15.2643375 

6.1534495 

.004291845 

234 

54756 

12812904 

15.2970585 

6.1622401 

.004273504 

235 

55225 

12977875 

15.3297097 

6.1710058 

.004255319 

236 

55696 

13144256 

15.3622915 

6.1797466 

.004237288 

237 

56169 

13312053 

15.3948043  i   6.1884628 

.004219409 

2:38 

56644 

13481272 

15.4272486     6.1971544 

.004201681 

239 

57121 

13651919 

15.4596248 

6.2058218 

.004184100 

240 

57600 

13824000 

15.4919334 

6.2144650 

.004166667 

241 

58081 

13997521 

15.5241747 

6.2230843 

.004149378 

242 

58564 

14172488 

15.5563492 

6.2316797 

.004132231 

243 

59049 

14348907     15.5884573 

6.2402515 

.004115226 

244 

59536 

14526784      15.6204994 

6.2487098 

.004098361 

245 

60025 

14706125     15.6524758 

6.2573248 

.004081633 

246 

60516 

14886936     15.6843871 

6.2658266 

.004065041 

247 

61009 

15069223 

15.7162336 

6.2743054 

.004048583 

248 

61504 

15252992 

15.7480157 

6.2827613 

.004032258 

TABLE   VIII. — SQUARES,    CUBES,    SQUARE   ROOTS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

249 

62001 

15438249 

15.7797338 

6.2911946 

.004016064 

250 

62500 

15625000 

15.8113883 

6.2996053 

.004000000 

251 

63001 

15813251 

15.8429795 

6.3079935 

.003984064 

252 

63504 

16003008 

15.8745079 

6.3163596 

.003968254 

253 

64009 

16194277 

15.9059737 

6.32-47035 

.003952569 

254 

64516 

16387064 

15.9373775 

6.3330256 

.003937008 

255 

65025 

16581375 

15.9687194 

6.3413257 

.003921569 

256 

65536 

16777216 

16.0000000 

6.3496042 

.003906250 

257 

66049 

16974593 

16.0312195 

6.3578611 

.003891051 

258 

66564 

17173512 

16.0623784 

6.3660968 

.003875969 

259 

67081 

17373979 

16.0934769 

6.3743111 

.003861004 

260 

67600 

17576000 

16.1245155 

6.3825043 

.003846154 

261 

68121 

17779581 

16.1554944 

6.3906765 

.003831418 

262 

68644 

17984723 

16.1864141 

6.3988279 

.003816794 

263 

69169 

18191447 

16.2172747 

6.4069585 

.003802281 

264 

69696 

18399744 

16.2480768 

6.4150687 

.003787879 

265 

70225 

18609625 

16.2788206 

6.4231583 

.003773585 

266 

70756 

18821096 

16.3095064 

6.4312276 

.003759398 

267 

71289 

19034163 

16.3401346 

6.4392767 

.003745318 

268 

71824 

19248832 

16.3707055 

6.4473057 

.003731343 

269 

72361 

19465109 

16.4012195 

6.4553148 

.003717472 

270 

72900 

19683000 

16.4316767 

6.4633041 

.003703704 

271 

73441 

19902511 

16.4620776 

6.4712736 

.003690037 

272 

73984 

20123648 

16.4924225 

6.4792236 

.003676471 

273 

7452J 

20346417 

16.5227116 

6.4871541 

.003663004 

274 

75076 

20570824 

16.5529454 

6.4950653 

.003649635 

275 

75625 

20796875 

16.5831240 

6.5029572 

.003636364 

276 

76176 

21024576 

16.6132477 

6.5108300 

.003623188 

277 

76729 

21253933 

16.6433170 

6.5186839 

.003610108 

278 

77284 

21484952 

16.6733320 

6.5265189 

.003597122 

279 

77841 

21717639 

16.7032931 

6.5343351 

.003584229 

280 

78400 

21952000 

16.7332005 

6.5421326 

.003571429 

281 

78961 

22188041 

16.7630546 

6.5499116 

.003558719 

282 

79524 

22425768 

16.7928556 

6.5576722 

.003546099 

283 

80089 

22665187 

16.8226038 

6.5654144 

.003533569 

284 

80656 

22906304 

16.8522995 

6.5731385 

.003521127 

285 

81225 

23149125 

16.8819430 

6.5808443 

.003508772 

286 

81796 

23393656 

16.9115345 

6.5885323 

.003496503 

287 

82369 

23639903 

16.9410743 

6.5962023 

.003484321 

288 

82944 

23887872 

16.9705627 

6.6038545 

.003472222 

289 

83521 

24137569 

17.0000000 

6.6114890 

.003460208 

290 

84100 

24389000 

17.0293864 

6.6191060 

.003448276 

291 

84681 

24642171 

17.0587221 

6.6267054 

.003436426 

292 

85264 

24897088 

17.0880075 

6.6342874 

.003424658 

293 

85849 

25153757 

17.1172428 

6.6418522 

.003412969 

294 

86436 

25412184 

17.1464282 

6.6493998 

.003401361 

295 

87025 

25672375 

17.1755640 

6.6569302 

.003389831 

296 

87616 

25934336 

17.2046505 

6.6644437 

.003378378 

297 

88209 

26198073 

17.2336879 

6.6719403 

.003367003 

298 

88804 

26463592 

17.2626765 

6.6794200 

.003355705 

299 

89401 

26730899 

17.2916165 

6.6868831 

.003344482 

300 

90000 

27000000 

17.3205081 

6.6913295 

.003333333 

301 

90601 

27270901 

17.3493516 

6.7017593 

.003322259 

302 

91204 

27543608 

17.3781472 

6.7091729 

.003311258 

303 

91809 

27818127 

17.4068952 

6.7165700 

.003300330 

304 

92416 

28094464 

17.4355958 

6.7239508 

.003289474 

305 

93025 

28372625 

17.4642492 

6.7313155 

.003278689 

306 

93636 

28652616 

17.4928557 

6.7386641 

.003267974 

307 

94249 

28934443 

17.5214155 

6.7459967 

.003257329 

308 

94864 

29218112 

17.5499288 

6.7533134 

.003246753 

309 

95481 

29503629 

17.5783958 

6.7606143 

.003236246 

310 

96100 

29791000 

17.6068169 

6.7678995 

.003225806 

64 


CUBE  ROOTS,  AND  RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

311 

96721 

30080231 

17.6351921 

6.7751690 

.003215434 

312 

97344 

30371328 

17.6635217 

6.7824229 

.003205128 

313 

97969 

30664297 

17.6918060 

6.7896613 

.003194888 

314 

98596 

30959144 

17.7200451 

6.7968844 

.003184713 

315 

99225 

31255875 

17.7482393 

6.8040921 

.003174603 

316 

99856 

31554496 

17.7763888 

6.8112847 

.003164557 

317 

100489 

31855013 

17.8044938 

6.8184620 

.003154574 

318 

101124 

32157432 

17.8325545 

6.8256242 

.003144654 

319 

101761 

32461759 

17.8605711 

6.8327714 

.003134796 

320 

102400 

32768000 

17.8885438 

6.8399037 

.003125000 

321 

103041 

33076161 

17.9164729 

6.8470213 

.003115265 

322 

103684 

33386248 

17.9443584 

6.8541240 

.003105590 

323 

104329 

33698267 

17.9722008 

6.8612120 

.003095975 

324 

104976 

34012224 

18.0000000 

6.8682855 

.003086420 

325 

105625 

34328125 

18.0277564 

6.8753443 

.003076923 

326 

106276 

34645976 

18.0554701 

6.8823888 

.003067485 

327 

106929 

34965783 

18.0831413 

6.8894188 

.003058104 

328 

107584 

35287552 

18.1107703 

6.8964345 

.003048780 

329 

108241 

35611289 

18.1383571 

6.9034359 

.003039514 

330 

108900 

35937000 

18.1659021 

6.9104232 

.003030303 

331 

109561 

36264691 

18.1934054 

6.9173964 

.003021148 

332 

110224 

30594368 

18.2208672 

6.9243556 

.003012048 

333 

110889 

36-J26037 

18.2482876 

6.9313008 

.003003003 

334 

111556 

37259704 

18.2756669 

6.9382321 

.002994012 

335 

112225 

37595375 

18.3030052 

6.9451496 

.002985075 

336 

112896 

37933056 

18.3303028 

6.9520533 

.002976190 

337 

113569 

38272753 

18.3575598 

6.9589434 

.002967359 

338 

114244 

38614472 

18.3847763 

6.9658198 

.002958580 

339 

114921 

38958219 

18.4119526 

6.9726826 

.002949853 

340 

115600 

39304000 

18.4390889 

6.9795321 

.002941176 

341 

116281 

39651821 

18.4661853 

6.9863681 

.002932551 

342 

116964 

40001688 

18.4932420 

6.9931906 

.002923977 

343 

117649 

40353607 

18.5202592 

7.0000000 

.002915452 

344 

118336 

40707584 

18.5472370 

7.0067962 

.002906977 

345 

119025 

41063625 

18.5741756 

7  0135791 

.002898551 

346 

119716 

41421736 

18.6010752 

7.0203490 

.002890173 

347 

120409 

417'81923 

18.6279360 

7.0271058 

.002881844 

348 

121104 

42144192 

18.6547581 

7.0338497 

.002873563 

349 

121801 

42508549 

18.6815417 

7.0405806 

.002865330 

350 

122500 

42875000 

18.7082869 

7.0472987 

.002857143 

351 

123201 

43243551 

18.7349940 

7.0540041 

.002849003 

352 

123904 

43614208 

18.7616630 

7.0606967 

.002840909 

353 

124609 

43986977 

18.7882942 

7.0673767 

.002832861 

354 

125316 

44361864 

18.8148877 

7.0740440 

.002824859 

355 

126025 

44738875 

18.8414437 

7.0806988 

.002816901 

356 

126736 

45118016 

18.8679623 

7.0873411 

.002808989 

357 

127449 

45499293 

18.8944436 

7.0939709 

.002801120 

358 

128164 

45882712 

18.9208879 

7.1005885 

.002793296 

359 

128881 

46268279 

18.9472953 

7.1071937 

.002785515 

360 

129600 

46656000 

18.9736660 

7.1137866 

.002777778 

361 

130321 

47045881 

19  0000000 

7.1203674 

.002770083 

362 

131044 

47437928 

19.0262976 

7.1269360 

.002762431 

363 

131769 

47832147 

19.0525589 

7.1334925 

.002754821 

364 

132496 

48228544 

19.0787840 

7.1400370 

.002747253 

365 

133225 

48627125 

19.1049732 

7.1465695 

.002739726 

366 

133956 

49027896 

19.1311265 

7.1530901 

.002732240 

367 

134689 

494:30863 

19.1572441 

7.1595988 

.002724796 

368 

1:35424 

498:36032 

19.1833261 

7.1660957 

.002717391 

369 

136161 

50243409 

19.2093727 

7.1725809 

.002710027 

370 

136900 

50653000 

19.2353841 

7.1790544 

.002702703 

371 

137641 

51064811 

19.2613603 

7.1855162 

.002695418 

372 

138384 

51478848 

19.2873015 

7.1919663 

.002688172 

TABLE   VIII. — SQUARES,,    CUBES,   SQUARE   ROOTS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

373 

139129 

51895117 

19.3132079 

7.1984050 

.002680965 

374 

139876 

52313624 

19.3390796 

7.2048322 

.002673797 

375 

140625 

52734375 

19.3649167 

7.2112479 

.002666667 

376 

141376 

53157376 

19.3907194 

7.2176522 

.002659574 

877 

142129 

53582633 

19.4164878 

7.2240450 

.002652520 

378 

142884 

54010152 

19.4422221 

7.2304268 

.002645503 

379 

143641 

54439939 

19.4679223 

7.2367972 

.002638522 

380 

144400 

54872000 

19.4935887 

7.2431565 

.002631579 

381 

145161 

55306341 

19.5192213 

7.2495045 

.002624672 

382 

145924 

55742968 

19.5448203 

7.2558415 

.002617801 

383 

146689 

56181887 

19.5703858 

7.2621675 

.002610966 

884 

147456 

56623104 

19.5959179 

7.2684824 

.002604167 

385 

148225 

57066625 

19.6214169 

7.2747864 

.002597403 

386 

148996 

57512456 

19.6468827 

7.2810794 

.002590674 

387 

149769 

57960603 

19.6723156 

7.2873617 

.002583979 

388 

150544 

58411072 

19.6977156 

7.2936330 

.002577'320 

389 

151321 

58863869 

19.7230829 

7.2998936 

.002570694 

390 

152100 

59319000 

19.7484177 

7.3061436 

.002564103 

391 

152881 

59776471 

19.7737199 

7.3123828 

.002557545 

392 

153664 

60236288 

19.7989899 

7.3186114 

.002551020 

393 

154449 

60698457 

19.8242276 

7.3248295 

.002544529 

394 

155236 

61162984 

19.8494332 

7.3310369 

.002538071 

395 

156025 

61629875 

19.8746069 

7.3372339 

.002531646 

396 

156816 

62099136 

19.8997487 

7.3434205 

.00*525253 

397 

157609 

62570773 

19.9248588 

7.3495966 

.002518892 

398 

158404 

63044792 

19.9499373 

7.3557624 

.002512563 

399 

159201 

63521199 

19.9749644 

7.3619178 

.002506266 

400 

160000 

64000000 

20.0000000 

7.3680630 

.002500000 

401 

160801 

64481201 

20.0249844 

7.3741979 

.002493766 

402 

161604 

64964808 

20.0499377 

7.3803227 

.002487562 

403 

162409 

65450827 

20.0748599 

7.3864373 

.002481390 

404 

163216 

65939264 

20.0997512 

7.3925418 

.002475248 

405 

164025 

66430125 

20.1246118 

7.3986363 

.002469136 

406 

164836 

66923416 

20.1494417 

7.4047206 

.002463054 

407 

165649 

67419143 

20.1742410 

7.4107950 

.002457002 

408 

166464 

67917'312 

20.1990099 

7.4168595 

.002450980 

409 

167'281 

68417929 

20.2237484 

7.4229142 

.002444988 

410 

168100 

68921000 

20.2484567 

7.4289589 

.002439024 

411 

168921 

69426531 

20.2731349 

7.4349938 

.002433090 

412 

169744 

69934528 

20.2977831 

7.4410189 

.002427184 

413 

170569 

70444997 

20.3224014 

7.4470342 

.002421308 

414 

171396 

70957944 

20.3469899 

7.4530399 

.002415459 

415 

172225 

71473375 

20.3715488 

7.4590359 

.002409639 

416 

173056 

71991296 

20.3960781 

7.4650223 

.002403846 

417 

173889 

72511713 

20.4205779 

7.4709991 

.002398082 

418 

174724 

73034632 

20.4450483 

7.4769664 

.002392344 

419 

175561 

73560059 

20.4694895 

7.4829242 

.002386635 

420 

176400 

74088000 

20.4939015 

7.4888724 

.002380952 

421 

177241 

74618461 

20.5182845 

7.4948113 

.002375297 

422 

178084 

75151448 

20.5426386 

7.5007406 

.002369668 

423 

178929 

•^5686967 

20.5669638 

7.5066607 

.002364066 

424 

179776 

,6225024 

20  5912603 

7.5125715 

.002358491 

425 

180625 

76765625 

20.6155281 

7.5184730 

.002352941 

426 

181476 

77308776 

20.6397674 

7.5243652 

.002347418 

427 

182329 

77854483 

20.6639783 

7.5302482 

.002341920 

428 

183184 

78402752 

20.6881609 

7.5361221 

.002336449 

429 

184041 

78953589 

20.7123152 

7.5419867 

.002331002 

430 

184900 

79507000 

20.7364414 

7.5478423 

.002325581 

431 

185761 

80062991 

20.7605395 

7.5536888 

.002320186 

432 

186624 

80621568 

20.7846097 

7.5595263 

.002314815 

433 

187489 

81182737 

20.8086520 

7.5653548 

.002309469 

434 

188356 

8174G504 

20.8326667 

7.5711743 

.002304147 

CUBE  HOOTS,  AND  RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

435 

189225 

82312875 

20.8566536 

7.5769849 

.002298851 

436 

190096 

82881856 

20.8806130 

7.5827865 

.002293578 

437 

190969 

83453453 

20.9045450 

7.5885793 

.002248330 

438 

191844 

84027672 

20.9284495 

7.5943633 

.002283105 

439 

192721 

84604519 

20.9523268 

7.6001385 

.002277904 

440 

193600 

85184000 

20.9761770 

7.6059049 

.002272727 

441 

194481 

85766121 

21.0000000 

7.6116626 

.002267574 

442 

195364 

863508S8 

21.0237960 

7.6174116 

.002262443 

443 

196249 

86938307 

21.0475652 

7.6231519 

.002257336 

444 

197136 

87528384 

21.0713075 

7.6288837 

.002252252 

445 

198025 

88121125 

21.0950231 

7.6:346067 

.002247191 

446 

198916 

88716536 

21.1187121 

7.6403213 

.002242152 

447 

199809 

89314623 

21.1423745 

7.6460272 

.002237136 

448 

200704 

89915392 

21  .  1660105 

7.6517247 

.002232143 

449 

201601 

90518849 

21  .  1896201 

7.  657413  j 

.002227171 

450 

202500 

91125000 

21.2132034 

7.6630943 

.002222222 

451 

20:3401 

91733851 

21.2367606 

7.66876G5 

.002217295 

452 

204304 

92345408 

21.2602916 

7.6744303 

.002212389 

453 

205209 

92959677 

21.2837967 

7.  -5800857 

.002207506 

454 

206116 

93576664 

21.3072758 

7.6857328 

.002202643 

455 

207025 

94196375 

21.3307290 

7.6913717 

.002197802 

456 

207936 

94818816 

21.3541565 

7.6970023 

.002192982 

457 

208849 

95443993 

21.3775583 

7.7026246 

.002188184 

458 

209764 

960;i!)12 

21.4009340 

7.7082:388 

.00218:3406 

459 

210681 

90702575) 

21.4242853 

7.7138448 

.002178649 

460 

211600 

97336000 

21.4476106 

7.7194426 

.002173913 

461 

212521 

97972181 

21.4709106 

7.7250325 

.002169197 

462 

213444 

9861  112S 

21.4941853 

7.7306141 

.002161502 

463 

214369 

99252847 

21.5174348 

7.7361877 

.002159827 

464 

215296 

99897344 

21.5406592 

7.7417'532 

.002155172 

465 

216225 

100544(52.-) 

21.5638587 

7.7473109 

.0021505:38 

466 

217150 

101194696 

21.5870331 

7.7528606 

.002145923 

467 

218089 

101847563 

21.6101828 

7.7584023 

.002141328 

468 

219024 

102503232 

21.6333077 

7.7639361 

.002136752 

469 

219961 

103161709 

21.6564078 

7.7094020 

.002132196 

470 

220900 

103823000 

21.6794834 

7.7749801 

.002127660 

471 

221841 

104487111 

21.7025344 

7.7804904 

.002123142 

472 

222784 

105154048 

21.7'255610 

7.7K>9928 

.002118644 

473 

223729 

105823817 

21.7485632 

7.7914875 

.002114165 

474 

224676 

106496424 

21.7715411 

7.7969745 

.002109705 

475 

225625 

107171875 

21.7944947 

7.8024538 

.002105263 

476 

226576 

107850176 

21.8174242 

7.8079254 

.002100840 

477 

227529 

108531333 

21.8403297 

7.81&3892 

.002096436 

478 

228484 

109215352 

21  8032111 

7.8188456 

.002092050 

479 

229441 

109902239 

21.8860686 

7.8242942 

.002087683 

480 

230400 

110592000 

21.9089023 

7.8297353 

.0020a8333 

481 

231361 

111284641 

21.9317122 

7.8351688 

.002079002 

482 

232324 

111980168 

21.9544984 

7.8405949 

.002074689 

483 

233289 

112678587 

21.9772610 

7.8460134 

.002070393 

484 

234256 

113379904 

2  .0000000 

7.8514244 

.002066116 

485 

235225 

114084125 

2  .0227155 

7.8568281 

.002061856 

486 

236196 

114791256 

2-  .0454077 

7.8622242 

.002057613 

487 

237169 

115501303 

2  .0680765 

7.8676130 

.002053:388 

488 

238144 

116214272 

2  .0907220 

7.8729944 

.002049180 

489 

239121 

116930169 

2  .1133441 

7  8783684 

.002044990 

490 

240100 

117649000 

22.1.359436 

7.8837352 

.002040816 

491 

241081 

118370771 

22.1585198 

7.8890946 

.002036660 

492 

242004 

119095488 

22.1810730 

7.8944468 

.002032520 

493 

243049 

119823157 

22.2036033 

7.8997917 

.002028398 

494 

244036 

120553784 

22.2261108 

7.9051294 

.002024291 

495 

245025 

121287375 

22.2485955 

7.9104599 

.002020202 

496 

24G016 

122023936 

22.2710575 

7.9157832 

.002010129 

TABLE   VIII. — SQUARES,   CUBES,   SQUARE    ROOTS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

407 

247009 

1227'63473 

22.2934968 

7.9210994 

.002012072 

498 

248004 

123505992 

22.3159136 

7.9264085 

.002008032 

499 

249001 

124251499 

22.3383079 

7.9317104. 

.002004008 

500 

250000 

125000000 

22.3606798 

7.9370053 

.002000000 

501 

251001 

125751501 

22.3830293 

7.9422931 

.001996008 

502 

252004 

126506008 

22.4053565 

7.9475739 

.001992032 

503 

253009 

127263527 

22.4276615 

7.9528477 

.001988072 

504 

254016 

128024064 

22.4499443 

7.9581144 

.001984127 

505 

255025 

128787625 

22.4722051 

7.9633743 

.001980198 

506 

256036 

129554216 

22.4944438 

7.9686271 

.001976285 

507 

257049 

130323843 

22.5166605 

7.9738731 

.001972387 

508 

258064 

131096512 

22.5388553 

7.9791122 

.001968504 

509 

259081 

131872229 

22.5610283 

7.9843444 

.001964637 

510 

260100 

132651000 

22.5831796 

7.9895697 

.001960784 

511 

261121 

133432831 

22.6053091 

7.9947883 

.001956947 

512 

262144 

134217728 

22.6274170 

8.0000000 

.001953125 

513 

263169 

135005697 

22.6495033 

8.0052049 

.001949318 

514 

264196 

135796744 

22.6715681 

8.0104032 

.001945525 

515 

265225 

136590875 

22.6936114 

8.0155946 

.001941748 

516 

266256 

137388096 

22.7156334 

8.0207794 

.001937984 

517 

267289 

138188413 

22.7376340 

8.0259574 

.001934236 

518 

268324 

138991832 

22.7596134 

8.0311287 

.001930502 

519 

269361 

139798359 

22.7815715 

8.0362935 

.001926782 

520 

270400 

140608000 

22.8035085 

8.0414515 

.001923077 

521 

271441 

141420761 

22.8254244 

8.0466030 

.001919386 

522 

272484 

142236648 

22.8473193 

8.0517479 

.001915709 

523 

273529 

143055667 

22.8691933 

8.0568862 

.001912046 

524 

274576 

143877824 

22.8910463 

8.0620180 

.001908397 

525 

275625 

144703125 

22.9128785 

8.0671432 

.001904762 

526 

276676 

145531576 

22.9346899 

8.0722620 

.001901141 

527 

277729 

146363183 

22.9564806 

8.0773743 

.001897533 

528 

278784 

147197952 

22.9782506 

8.0824800 

.001893939 

529 

279841 

148035889 

23.0000000 

8.0875794 

.001890359 

530 

280900 

148877000 

23.0217289 

8.0926723 

.001886792 

531 

281961 

149721291 

23.0434372 

8.0977589 

.001883239 

532 

283024 

150568768 

23.0651252 

8.1028390 

.001879699 

533 

284089 

151419437 

23.0867928 

8.1079128 

.001876173 

534 

285156 

152273304 

23.1084400 

8.1129803 

.001872659 

535 

286225 

153130375 

23.1300670 

8.1180414 

.001869159 

536 

287296 

153990656 

23.1516738 

8.1230962 

.001865672 

537 

288369 

154854153 

23.1732605 

8.1281447 

.001862197 

538 

289444 

155720872 

23.1948270 

8.1:331870 

.001858736 

539 

290521 

156590819 

23.2163735 

8.1382230 

.00.1855288 

540 

291600 

157464000 

23.2379001 

8.1432529 

.001851852 

541 

292681 

158340421 

23.2594067 

8.1482765 

.001848-429 

542 

293764 

159220088 

23.2808935 

8.1532939 

.001845018 

543 

294849 

160103007 

23.3023604 

8.1583051 

.001841621 

544 

295936 

160989184 

23.3238076 

8.1633102 

.001838235 

545 

297025 

161878625 

23.3452351 

8.1683092 

.001834862 

546 

298116 

162771336 

23.3666429 

8.1733020 

.001831502 

547 

299209 

163667323 

23.3880311 

8.1782888 

.001828154 

548 

300304 

164566592 

23.4093998 

8.1832695 

.001824818 

549 

301401 

165469149 

23.4307490 

8.1882441 

.001821494 

550 

302500 

166375000 

23.4520788 

8.1932127 

.001818182 

551 

303601 

167284151 

23.4733892 

8.1981753 

.001814882 

552 

304704 

168196606 

23.4946802 

8.2031319 

.001811594 

553 

305809 

169112377 

23.5159520 

8.2080825 

.001808318 

554 

306916 

170031464 

23.5372046 

8.2130271 

.001805054 

555 

308025 

170953875 

23.5584380 

8.2179657 

.001801802 

556 

309136 

171879616 

23.5796522 

8.2228985 

.001798561 

557 

310249 

172808693 

23.6008474 

8.2278254 

.001795332 

558 

311364 

173741112 

23.6220236 

8.2327463 

.001792115 

CUBE  ROOTS,  AND  RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

559 

312481 

174676879 

23.6431808 

8.2376614 

001788909 

560 

313000 

175010000 

23.6643191 

8.2425706 

.001785714 

501 

314721 

170558481 

23.6854386 

8.2474740 

.001782531 

503 

315844 

177504328 

23.7065392 

8.2523715 

001779359 

503 

316969 

178453547 

23.7276210 

8.2572633 

001770199 

504 

318096 

179400144 

23.7486842 

8.2021492 

.001773050 

505 

319225 

180302125 

23.7097280 

8.2670294 

.001709912 

506 

320356 

181321490 

23.7907545 

8.2719039 

.001766784 

507 

321489 

182284203 

23.8117618 

8.2767726 

001763008 

508 

322624 

183250432 

23.8327506 

8.2816355 

.001760503 

509 

323701 

184220000 

S&.  8537209 

8.2864928 

.001757469 

570 

324900 

185193000 

23.8746728 

8.2913444 

.001754386 

571 

326041 

180109411 

83.8956063 

8.2961903 

.001751313 

572 

327184 

187149248 

23.9165215 

8.3010304 

.001748252 

573 

328329 

188132517 

23.9374184 

8.3058651 

.001745201 

574 

329476 

189119224 

23.9582971 

8.3106941 

.001742160 

575 

330625 

190109375 

23.9791576 

8.3155175 

.001739130 

576 

331776 

191102976 

24.0000000 

8.3203353 

.001736111 

577 

332929 

192100033 

24.0208243 

8.3251475 

.001733102 

578 

334084 

193100552 

24.0410306 

8.3299542 

.001730104 

579 

335241 

194104539 

24.0024188 

8.3347553 

.001727116 

580 

336400 

195112000 

24.0831891 

8.3395509 

.001724138 

581 

337501 

196122941 

24.1039416 

8.3443410 

.001721170 

582 

338724 

197137368 

24.1246702 

8.3491256 

.001718213 

583 

339889 

198155287 

24.1453929 

8.3539047 

.001715266 

584 

341056 

199176704 

24.1060919 

8.3586784 

.001712329 

585 

342225 

200201625 

24.1867732 

8.3034466 

.001709402 

586 

343396 

201230056 

24.2074369 

8.3682095 

001706485 

587 

344569 

202262003 

24.2280829 

8.3729668 

.001703578 

588 

345744 

203297472 

24.2487113 

8.3777188 

.001700680 

589 

340921 

204330409 

24.2693222 

8.3824653 

.001697793 

590 

348100 

205379000 

24.2899156 

8.3872065 

.001694915 

591 

349281 

206425071 

24.3104916 

8.3919423 

.001692047 

592 

350404 

207474688 

24.3310501 

8.3966729 

.001689189 

593 

351649 

208527857 

24.3515913 

8.4013981 

.001086341 

594 

352836 

2095S4584 

24.3721152 

8.4061180 

.001683502 

595 

354025 

210644875 

24.3926218 

8.4108326 

001680672 

596 

355216 

211708736 

24.4131112 

8.4155419 

.001677852 

597 

356409 

212776173 

24.4335834 

8.4202460 

.001675042 

598 

357004 

213847192 

24.4540385 

8.4249448 

.001672241 

599 

358801 

214921799 

24.4744765 

8.4296383 

.001009449 

600 

360000 

216000000 

24.4948974 

8.4348267 

.001666667 

601 

361201 

217081801 

24.5153013 

8.4390098 

.001603894 

602 

302404 

218167203 

24.5356883 

8.4436877 

.001001130 

603 

303009 

219256227 

24.5560583 

8.4483605 

.001058375 

604 

364816 

220348864 

24.5764115 

8.4530281 

.001055629 

605 

300025 

221445125 

24.5967478 

8.4576906 

.001652893 

606 

307236 

222545016 

24.6170673 

8.4623479 

.001650165 

607 

368449 

223648543 

24.6373/00 

8.4670001 

.001647446 

608 

309064 

224755712 

24.6576560 

8.4710471 

.001644737 

609 

370881 

225860529 

24.6779254 

8.4762892 

.001642036 

610 

372100 

226981000 

24.6981781 

8.4809261 

.001639344 

611 

373321 

228099131 

24.7184142 

8.4855579 

.001030061 

612 

374544 

229220928 

24.7386338 

8.4901848 

001033987 

613 

375769 

230346397 

24.7588368 

8.4948065 

001031321 

614 

376996 

231475544 

24.7790234 

8.4994233 

.001628664 

615 

378225 

232608375 

24.7991935 

8.5040350 

.001626010 

616 

379456 

233744896 

24.8193473 

8.5086417 

.001023377 

617 

380689 

234885113 

24.8394847 

8.5132435 

.001620746 

618 

381924 

236029032 

24.8596058 

8.5178403 

.001618123 

619 

383161 

237176659  1   24.8797106 

8.5224321 

.001015509 

620 

384400 

238328000    24.8997992  j   8.5270189 

.001612903 

TABLE   VIII. — SQUARES,    CUBES,    SQ.UARE    ROOTS. 


No. 

Squares 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

621 

385641 

239483061 

24.9198716 

8.5316009 

.001610306 

622 

386884 

240641848 

24.9399278 

8.5361780 

.001607717 

623 

388129 

241804367 

24.9599679 

8.5407501 

.001605136 

624 

389376 

242970624 

24.9799920 

8.5453173 

.001602564 

625 

390625 

244140625 

25.0000000 

8.5498797 

.001600000 

62(3 

391876 

245314376 

25.0199920 

8.;j544372 

.001597444 

627    393129 

246491883 

25.0399681 

8.5589899 

.001594896 

628    394384 

247673152 

25.0599282 

8.5635377 

.001592357 

629    395641 

•  248858189 

25.07987'24 

8.5680807 

.001589825 

630 

396900 

250047000 

25.0998008 

8.5726189 

.001587302 

631 

398161 

251239591 

25.1197134 

8.5771523 

.001584786 

632 

399424 

252435968 

25.1396102 

8.5816809 

.001582278 

633 

400689 

253636137 

25.1594913 

8.5862047 

.001579779 

634 

401956 

254840104 

25.1793566 

8.5907238 

.001577287 

635 

403225 

256047875 

25.1992063 

8.5952380 

.001574803 

636 

404496 

257259456 

25.2190404 

8.5997476 

.001572327 

637 

405769 

25&4748S3 

25.2388589 

8.6042525 

.001569859 

638 

407044 

259694072 

25.2586619 

8.6087526 

.001567398 

639 

408321 

260917119 

25.2784493 

8.6132480 

.001564945 

640 

409600 

262144000 

25.2982213 

8.6177388 

.001562500 

641 

410881 

263374721 

25.3179778 

8.6222248 

.001560062 

642 

412164 

264609288 

25.3377189 

8.6267063 

.001557632 

643 

413449 

265847707 

25.3574447 

8.6311830 

.001555210 

644 

414736 

267089984 

25.3771551 

8.6356551 

.001552795 

645    416025 

26833G125 

25.3968502 

8.6401226 

.001550388 

646 

417316 

269586136 

25.4165301 

8.6445855 

.001547988 

647 

418609 

270840023 

25.4361947 

8.6490437 

.001545595 

648  i  419904 

272097792 

25.4558441 

8.6534974 

.001543210 

649 

421201 

273359449 

25.4754784 

8.6579465 

.001540832 

650 

422500 

274625000 

25.4950976 

8.6623911 

.001538462 

651 

423801 

275894451 

25.5147016 

8.6668310 

.001536098 

652    425104 

277167808 

25.5342907 

8.6712665 

.001533742 

653  j  426409 

278445077 

25.5538647 

8.6756974 

.001531394 

654 

427716 

279726264 

25.5734237 

8.6801237 

.001529052 

655 

429025 

281011375 

25.5929678 

8.6845456 

.001526718 

656 

430336 

282300416 

25.6124969 

8.6889630 

.001524390 

657 

431649 

283593393 

25.6320112 

8.6933759 

.001522070 

658 

432964 

284890312 

25.6515107 

8.6977843 

.001519757 

659 

434281 

286191179 

25.6709953 

8.7021882 

.001517451 

660 

435600 

287496000 

25.6904652 

8.7065877 

.001515152 

661 

436921 

288804781 

25.7099203 

8.7109827 

.001512859 

662 

438244 

290117528 

25.7293607 

8.7153734 

.001510574 

663 

439569 

291434247 

25.7487864 

8.7197596 

.001508296 

664 

440896 

292754944 

25.7681975 

8.7241414 

.001506024 

665 

442225 

294079625 

25.7875939 

8.7285187 

.001503759 

666 

443556 

295408296 

25.8069758 

8.7328918 

.001501502  • 

667 

444889 

296740963 

25.8263431 

8.7372604 

.001499250 

668 

446224 

298077632 

25.8456960 

8.7416246 

.001497006 

6S9 

447561 

299418309 

25.8650343 

8.7459846 

.001494768 

670 

448900 

300763000 

25.8843582 

8.7503401 

.001492537 

671 

450241 

302111711 

25.9036677 

8.7546913 

.001490313 

672 

451584 

303464448 

25.9229628 

8.7590383 

.001488095 

673 

452929 

304821217 

25.9422435 

8.7633809 

.001485884 

674 

454276 

306182024 

25.9615100 

8.7677192 

.001483680 

675 

455625 

307546875 

25.9807621 

8.7720532 

.001481481 

676 

456976 

308915776 

26.0000000 

8.7763830 

.001479290 

677 

458329 

310288733 

26.0192237 

8.7807084 

.001477105 

678 

459684 

311665752 

26.0384331 

8.  7850296 

.001474926 

679 

461041 

313046839 

T6.  0576284 

8.7893466 

.001472754 

680 

462400 

314432000 

26.0768096 

8.7936593 

.001470588 

681 

463761 

315821241 

26.0959767 

8.7979679 

.001468429 

682 

465124 

317214568 

26.1151297 

8.8022721 

.001466276 

CUBE  ROOTS,  AND  RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Hoots. 

Cube  Roots. 

Reciprocals. 

683 

466489 

318611987 

26.1342687 

8.8065722 

.001464129 

684 

467856 

320013504 

26.1533937 

8.8108681 

.001401988 

685 

469225 

321419125 

20.1725047 

8.8151598 

.001459854 

686 

470596 

322828856 

26.1916017 

8.8194474 

.001457720 

687 

471969 

324242703 

26.2106848 

8.8237307 

.001455604 

688 

473344 

325660672 

26.2297541 

8.8280099 

.001453488 

689 

474721 

327082769 

26.2488095 

8.8322850 

.001451379 

690 

476100 

328509000 

26.2678511 

8.8365559 

.001449275 

691 

477481 

329939371 

2(5.2868789 

8.8408227 

.001447178 

692 

478864 

331373888 

26.3058929 

8.8450854 

.001445087 

693 

480249 

332812557 

26.3248932 

8.8493440 

.001443001 

694 

481636 

334255384 

26.3438797 

8.8535985 

.001440922 

695 

483025 

335702375 

26.3628527 

8.8578489 

.001438849 

696 

484416 

337153530 

26.3818119 

8.8620952 

.001436782 

697 

485809 

338608873 

26.4007576 

8.8663375 

.001434720 

698 

487204 

340068392 

26.419(5890 

8.8705757 

.001432065 

699 

488001 

341532099 

20.4:380081 

8.8748099 

.001430015 

700 

490000 

343000000 

20.4575131 

8.8790400 

.001428571 

7'01 

491401 

344472101 

20.4704040 

8.8832001 

.001426534 

702 

492804 

345948408 

20.4952820 

8.8874882 

.001424501 

703 

494209 

34742892? 

20.5141472 

8.8917063 

.001422475 

704 

495616 

348913664 

20.5329983 

8.8959204 

.001420455 

705 

497025 

350402625 

26.5518361 

8.9001304 

.001418440 

706 

498436 

351895816 

20.5706605 

8.9043306 

.001416431 

707 

499849 

353393243 

26.5894716 

8.9085387 

.001414427 

708 

501264 

354894912 

26.0082094 

8.9127369 

.001412429 

709 

502G81 

356400829 

26  0270539 

8.9109311 

.001410437 

710 

504100 

357911000 

26.6458252 

8.9211214 

.001408451 

711 

505521 

359425431 

26.6645833 

8.9253078 

.001400470 

712 

506944 

360944128 

26.6833281 

8.9294902 

.001404494 

713 

508369 

362467097 

26.7020598 

8.9336687 

.001402525 

714 

509796 

363994344 

26.7207784 

8.9378433 

.001400560 

715 

511225 

365525875 

26.7394839 

8.9420140 

.001398601 

716 

512656 

367061696 

26.7581763 

8.9461809 

.001396648 

717 

514089 

368601813 

26.7768557 

8.9503438 

001394700 

718 

515524 

370146232 

26.7955220 

8.9545029 

.001392758 

719 

516961 

371694959 

26.8141754 

8.9586581 

.001390821 

720 

518400 

373248000 

26.8328157 

8.9628095 

.001388889 

721 

519841 

374805361 

20.8514432 

8.9669570 

.001386963 

722 

521284 

376367048 

26.  8700577 

8.9711007 

.001385042 

723 

522729 

37793:3067 

26.8886593 

8.9752406 

.001383126 

724 

524176 

379503424 

26.9072481 

8.9793766 

.001381215 

725 

525625 

381078125 

26.9258240 

8.9835089 

.001379310 

726 

527076 

382657176 

26.9443872 

8.9876373 

.001377410 

727 

528529 

384240583 

20.9029375 

8.9917620 

.001375516 

728 

529984 

385828352 

26.9814751 

8.9958829 

.001373626 

729 

531441 

387420489 

27.0000000 

9.0000000 

.001371742 

730 

532900 

389017000 

27.0185122 

9.0041134 

.001369863 

731 

534361 

390617891 

27.0370117 

9.0082229 

.001367989 

732 

535824 

392223168 

27.0554985 

9.0123288 

.001306120 

733 

537'289 

393832837 

27.0739727 

9.0164309 

.001364256 

734 

538756 

395446904 

27.0924344 

9.0205293 

.001362398 

735 

540225 

397065375 

27.1108834 

9.0246239 

.001360544 

736 

541696 

398688256 

27.1293199 

9.0287149 

.001358696 

737 

543169 

400315553 

27.1477439 

9.0328021 

.001356852 

738 

544644 

401947272 

27.1661554 

9.0368857 

.001355014 

739 

546121 

403583419 

27.1845544 

9.0409655 

.001353180 

740 

547600 

405224000 

27.2029410 

9.0450419 

.001351351 

741 

549081 

406869021 

27.2213152 

9.0491142 

.001349528 

742 

550564 

408518488 

27.2390709 

9.0531831 

.001347709 

743 

552049 

410172407 

27.2580263 

9.0572482 

.001345895 

744 

553536 

411830784 

27.2703034 

9.0613098 

.001344086 

71 


TABLE    VIII. — SQUARES,    CUBES,    SQUARE    ROOTS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

745 

555025 

413493625 

27.2946881 

9.0653677 

.001342282 

746 

556516 

415160936 

27.3130006 

9.0694220 

.001340483 

747 

558009 

416832723 

27.3313007 

9.0734726 

.001338688 

748 

559504 

418508992 

27.3495887 

9.0775197 

.001336898 

749 

561001 

420189749 

27.3678644 

9.0815631 

.001335113 

750 

562500 

421875000 

27.3861279 

9.0856030 

.001333333 

751 

564001 

423564751 

27.4043792 

9.0896392 

.001331558 

752 

565504 

425259008 

27.4226184 

9.0936719 

.001329787 

753 

567009 

426957777 

27.4408455 

9.0977010 

.001328021 

754 

568516 

428661064 

27.4590604 

9.1017265 

.001326260 

755 

570025 

430368875 

27.4772633 

9.1057485 

.001324503 

756 

571536 

432081216 

27.4954542 

9.1097669 

.001322751 

757 

573049 

433798093 

27.5136330 

9.1137818 

.001321004 

758 

574564 

435519512 

27.5317998 

9.1177931 

.001319261 

759 

576081 

437245479 

27.5499546 

9.1218010 

.001317523 

760 

577600 

438976000 

27.5680975 

9.1258053 

.001315789 

761 

579121 

440711081 

27.5862284 

9.1298061 

.001314060 

762 

580644 

442450728 

27.6043475 

9.1338034 

.001312336 

763 

582169 

444194947 

27.62.^4546 

9.1377971 

.001310616 

764 

583696 

445943744 

27.6405499 

9.1417874 

.001308901 

765 

585225 

447697125 

27.6586334 

9.1457742 

.001307190 

766 

586756 

449455096 

27.6767050 

9.1497576 

.001305483 

767 

588289 

451217663 

27.6947648 

9.1537375 

.001303781 

768 

589824 

452984832 

27.7128129 

9.1577139 

.001302083 

769 

591361 

454756609 

27.7308492 

9.1616869 

.001300390 

770 

592900 

456533000 

27.7488739 

9.1656565 

.001298701 

771 

594441 

458314011 

27.7668868 

9.1696225 

.001297017 

772 

595984 

460099648 

27.7848880 

9.1735852 

.001295337 

773 

597529 

461889917 

27.8028775 

9.1775445 

.001293661 

774 

599076 

463684824 

27.8208555 

9.1815003 

.001291990 

775 

600625 

465484375 

27.8388218 

9.1854527 

.001290323 

776 

602176 

467288576 

27.8567766 

9.1894018 

.001288660 

777 

603729 

469097433 

27.8747197 

9.1933474 

.001287001 

778 

605284 

470910952 

27.8926514 

9.1972897 

.001285347 

779 

606841 

472729139 

27.9105715 

9.2012286 

.001283697 

780 

608400 

474552000 

27.9284801 

9.2051641 

.001282051 

781 

609961 

476379541 

27.9463772 

9.2090962 

.001280410 

782 

611524 

478211768 

27.9642629 

9.2130250 

.001278772 

783 

613089 

480048687 

27.9821372 

9.2169505 

.001277139 

784 

614656 

481890304 

28.0000000 

9.2208726 

.001275510 

785 

616225 

483736625 

28.0178515 

9.2247914 

.001273885 

786 

617796 

485587656 

28.0356915 

9.2287068 

.001272265 

787 

619369 

487443403 

28.0535203 

9.2326189 

.001270648 

788 

620944 

489303872 

28.0713377 

9.2365277 

.001269036 

789 

622521 

491169069 

28.0891438 

9.2404333 

.001267427 

790 

624100 

493039000 

28.1069386 

9.2443355 

.001265823 

791 

625681 

494913671 

28.1247222 

9.2482344 

.001264223 

792 

627264 

496793088 

28.1424946 

9.2521300 

.001262626 

793 

628849 

498677257 

28.1602557 

9.2560224 

.001261034 

794 

630436 

500566184 

28.1780056 

9.2599114 

.001259446 

795 

632025 

502459875 

28.1957444 

9.2637973 

.001257862 

796 

633616 

504358336 

28.2134720 

9.2676798 

.001256281 

797 

635209 

506261573 

28.2311884 

9.2715592 

.001254705 

798 

636804 

508169592 

28.2488938 

9.2754352 

.001253133 

799 

638401 

510082399 

28.2665881 

9.2793081 

.001251564 

800 

640000 

512000000 

28.2842712 

9.2831777 

.001250000 

801 

641601 

513922401 

28.3019434 

9.2870440 

.001248439 

802 

643204 

515849608 

28.3196045 

9.2909072 

.001246883 

803 

644809 

517781627 

28.  £372546 

9.2947671 

.001245330 

804 

646416 

519718464 

28.3548938 

9.2986239 

.001243781 

805 

648025 

521660125 

28.3725219 

9.3024775 

.001242236 

806 

649636 

523606616    28.3901391 

9.3063278 

.001240695 

CUBE  ROOTS,  AND  RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

807 

651249 

525557943 

28.4077454 

9.3101750 

.001239157 

808 

652864 

527514112 

28.4253408 

9.3140190 

.001237624 

809 

654481 

529475129 

28.4429253 

9.3178599 

.001236094 

810 

656100 

531441000 

28.4604989 

9.3216975 

.001234568 

811 

657721 

533411731 

28.4780617 

9.3255320 

.001233046 

812 

659344 

535387328 

28.4956137 

9.3293634 

.001231527 

813 

660969 

537367797 

28.5131549 

9.3331916 

.001230012 

814 

662596 

539353144 

28.5306852 

9.3370167 

.001228501 

815 

664225 

541343375 

28.5482048 

9.3408386 

.001226994 

816 

665856 

543338496 

28.5657137 

9.3446575 

.001225490 

817 

667489 

545338513 

28.5832119 

9.3484731 

.001223990 

818 

669124 

547343432 

28.6006993 

9.3522857 

.001222494 

819 

670761 

549353259 

28.6181760 

9.  &  60952 

.001221001 

820 

672400 

551368000 

28.6356421 

9.3599016 

.001219512 

821 

674041 

553387661 

28.6530976 

9.3637049 

.001218027 

822 

675684 

555412248 

28.6705424 

9.3675051 

.001216545 

823 

677329 

557441767 

28.6879766 

9.3713022 

.001215067 

824 

678976 

559476224 

28.7054002 

9.3750963 

.001213592 

825 

680625 

561515625 

28.7228132 

9.3788873 

.001212121 

826 

682276 

563559976 

28.7402157 

9.3826752 

.001210654 

827 

683929 

565609283 

28.7576077 

9.3864600 

.001209190 

828 

685584 

567663552 

28.7749891 

9.3902419 

.001207729 

829 

687241 

569722789 

28.7923601 

9.3940206 

.001206273 

830 

688900 

571787000 

28.8097206 

9.3977964 

.001204819 

831 

690561 

573856191 

28.8270706 

9.4015691 

.001203369 

832 

692224 

575930368 

28.8444102 

9.4053387 

.001201923 

833 

693889 

578009537 

28.8617394 

9.4091054 

.001200480 

834 

695556 

580093704 

28.8790582 

9.4128690 

.001199041 

835 

697225 

582182875 

28.8963666 

9.4166297 

.001197605 

836 

698896 

584277056 

28.9136646 

9.4203873 

.001196172 

837 

700569 

586376253 

28.9309523 

9.4241420 

.001194743 

838 

702244 

588480472 

28.9482297 

9.4278936 

.001193317 

839 

703921 

590589719 

28.9654967 

9.4316423 

.001191895 

840 

705600 

592704000 

28.9827535 

9.4353880 

.001190476 

841 

707281 

594823321 

29.0000000 

9.4391307 

.001189061 

842 

708964 

596947688 

29.0172363 

9.4428704 

.001187648 

843 

710649 

599077107 

29.0344623 

9.4466072 

.001186240 

844 

712336 

601211584 

29.0516781 

9.450,3410 

.001184834 

845 

714025 

603351125 

29.0688837 

9.4540719 

.001183432 

846 

715716 

605495736 

29.0860791 

9.4577999 

.001182033 

847 

717409 

607645423 

29.1032644 

9.4615249 

.001180638 

848 

719104 

609800192 

29.1204396 

9.4652470 

.001179245 

849 

720801 

611960049 

29.1376046 

9.4689661 

.001177856 

850 

722500 

614125000 

29.1547595 

9.4726824 

.001176471 

851 

724201 

616295051 

29.1719043 

9.4763957 

.001175088 

852 

725904 

618470208 

29.1890390 

9.4801061 

.001173709 

853 

727609 

620650477 

29.2061637 

9.4838136 

.001172333 

854 

729316 

622835864 

29.2232784 

9.4875182 

.001170960 

855 

731025 

625026375 

29.2403830 

9.4912200 

.001169591 

856 

732736 

627222016 

29.2574777 

9.4949188 

.001168224 

857 

734449 

629422793 

29.2745623 

9.4986147 

.001166861 

858 

736164 

631628712 

29.2916370 

9.5023078 

.001165501 

859 

737881 

633839779 

29.3087018 

9.5059980 

.001164144 

860 

739600 

636056000 

29.3257566 

9.5096854 

.001162791 

861 

741321 

638277381 

29.3428015 

9.5ia3699 

.001161440 

862 

743044 

640503928 

29.  £598365 

9.5170515 

.001160093 

863 

744769 

642735647 

29.3768616 

9.5207303 

.001158749 

864 

746496 

644972544 

29.3938769 

9.5244063 

.001157407 

865 

748225 

647214625 

29.4108823 

9.5280794 

.001156069 

866 

749956 

649461896 

29.4278779 

9.5317497 

.001154734 

867 

751689 

651714363 

29.4448637 

9.5354172 

.001153403 

868 

753424 

653972032 

29.4618397 

9.5390818 

,001152074 

TABLE    VIII. —  SQUARES,    CUBES,    SQUARE    ROOTS. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

869 

755161 

656234909 

29.4788059 

9.5427437    .001150748 

870 

756900 

658503000 

29.4957624 

9.5464027 

.001149425 

871 

758641 

660776311 

29.5127091 

9.5500589 

.001148106 

872 

760384 

663054848 

29.5296461 

9.5537123 

.001146789 

873 

762129 

665338617 

29.5465734 

9.5573630 

.001145475 

874 

763876 

667627624 

29.5634910 

9.5610108 

.001144165 

875 

765625 

669921875 

29.5803989 

9.5646559 

.001142857 

876 

767376 

672221376    29.5972972 

9.5682982 

.001141553 

877 

769129 

674526133  |  29.6141858 

9.5719377 

.001140251 

878 

770884 

676836152 

29.6310648 

9.5755745 

.001138952 

879 

772641 

679151439 

29.6479342 

9.5792085 

.001137656 

880 

774400 

681472000 

29.6647939 

9.5828397 

.001136364 

881 

776161 

683797841 

29.6816442 

9.5864682 

.001135074 

882 

777924 

686128968 

29.6984848 

9.5900939 

.001ia3787 

883 

779689 

688465387 

29.7153159 

9.5937169 

.001132503 

884 

781456 

690807104 

29.7321375 

9.5973873 

.001131222 

885 

783225 

693154125 

29.7489496 

9.6009548 

.001129944 

886 

784996 

695506456 

29.7657'521 

9.6045696 

.001128668 

887 

786769 

697864103 

29.7825452 

9.6081817 

.001127396 

888 

788544 

700227072 

29.7993289 

9.6117911 

.001126126 

889 

790321 

702595369 

29.8161030 

9.6153977 

.001124859 

890 

792100 

704969000 

29.8328678 

9.6190017 

.001123596 

891 

793881 

707347971 

29.8496231 

9.6226030 

.001122334 

892 

795664 

709732288 

29.8663690 

9.6262016 

.001121076 

893 

797449 

712121957 

29.8831056 

9.6297975 

.001119821 

894 

799236 

714516984 

29.8998328 

9.6333907 

.001118568 

895 

801025 

716917375 

29.9165506 

9.6369812 

.001117318 

896 

802816 

719323136 

29.9332591 

9.6405690 

.001116071 

897 

804609 

721734273 

29.9499583 

9.6441542 

.001114827 

898 

806404 

724150792 

29.9666481 

9.6477367 

.001113586 

899 

808201 

726572699 

29.9833287 

9.6513166 

.001112347 

900 

810000 

729000000 

30.0000000 

9.6548938 

.001111111 

901 

811801 

731432701 

30.0166620 

9.6584684 

.001109878 

902 

813604 

733870808 

30.0333148 

9.6620403 

.001108647 

903 

815409 

736314327 

30.0499584 

9.6656096 

.001107420 

904 

817216 

738763264 

30.0665928 

9.6691762 

.001106195 

905 

819025 

741217625 

30.0832179 

9.6727403 

.001104972 

906 

820836 

743677416 

30.0998339 

9.6763017 

.001103753 

907 

822649 

746142643 

30.1164407 

9.6798604 

.001102536 

908 

824464 

748613312 

30.1330383 

9.6834166 

.001101322 

909 

826281 

751089429 

30.1496269 

9.6869701 

.001100110 

910 

828100 

753571000 

30.1662063 

9.6905211 

.001098901 

911 

829921 

756058031 

30.1827765 

9.6940694 

.001097695 

912 

831744 

758550528 

30.1993377 

9.6976151 

.001096491 

913 

833569 

761048497 

30.2158899 

9.7011583 

.001095290 

914 

835396 

763551944 

30.2324329 

9.7046989 

.001094092 

915 

837225 

766060875 

30.2489669 

9.7082369 

.001092896 

916 

839056 

768575296 

30.2654919 

9.7117723 

.001091703 

917 

840889 

771095213 

30.2820079 

9.7153051 

.001090513 

918 

842724 

773620632 

30.2985148 

9.7188354 

.001089326 

919 

844561 

776151559 

30.3150128 

9.7223631 

.001088139 

920 

846400 

778688000 

30.3315018 

9.7258883 

.001086957 

921 

848241 

781229961 

30.3479818 

9.7294109 

.001085776 

922 

850084 

783777448 

30.3644529 

9.7329309 

.001084599 

923 

851929 

786&30467 

30.3809151 

9.7364484 

.001083423 

924 

853776 

788889024 

30.3973683 

9.7399634 

.001082251 

925 

855625 

791453125 

30.4138127 

9.7434758 

.001081081 

926 

857476 

794022776 

30.4302481 

9.7469857 

.001079914 

927 

859329 

796597983 

30.4466747 

9.7504930 

.001078749 

928 

861184 

799178752 

30.4630924 

9.7539979 

.001077586 

929 

863041 

801765089 

30.4795013 

9.7575002 

.001076426 

930 

864900 

804357000 

30.4959014 

9.7610001 

.001075269 

74 


CUBE  ROOTS,  AND  RECIPROCALS. 


No. 

Squares. 

Cubes. 

Square 
Hoots. 

Cube  Roots. 

Reciprocals. 

931 

866761 

806954491 

30.5122926 

9.7644974 

.001074114 

932 

868624 

809557568 

30.5286750 

9.7679922 

.001072961 

933 

870489 

812166237 

30.5450487 

9.7714845 

.001071811 

934 

872356 

814780504 

30.5614136 

9.  7749743 

.001070664 

935 

874225 

817400375 

30.5777697 

9.7784616 

.001069519 

936 

876096 

820025856 

30.5941171 

9.7819466 

.001068376 

937 

877969 

822656953 

30.6104557 

9.7854288 

.001067236 

938 

879844 

825293672 

30.6267857 

9.7889087 

.001066098 

839 

881721 

827936019 

30.6431069 

9.7923861 

.001064963 

940 

883600 

830584000 

30.6594194 

9.7958611 

.001063830 

941 

885481 

833237621 

30.6757233 

9.7993336 

.001062699 

942 

887364 

835896888 

30.6920185 

9.8028036 

.001061571 

943 

889249 

838561807 

30.7083051 

9.8062711 

.001060445 

944 

891136 

841232384 

hO.  7245830 

9.8097362 

.001059322 

945 

893025 

843908625 

30.7408523 

9.8131989 

.001058201 

946 

894916 

846590536 

30.7571130 

9.8166591 

.001057082 

947 

896809 

849278123 

30.7733651 

9.8201169 

.001055966 

948 

898704 

851971392 

30.7896086 

9.8235723 

.001054852 

949 

900601 

854670349 

30.8058436 

9.8270252 

.001053741 

950 

902500 

857375000 

30.8220700 

9.8304757 

.001052632 

951 

904401 

860085351 

30.8382879 

9.8339238 

.001051525 

952 

906304 

862801408 

30.8544972 

9.8373695 

.001050420 

953 

908209 

865523177 

30.8706981 

9.8408127 

.001049318 

954 

910116 

868250664 

30.8868904 

9.8442536 

.001048218 

955 

912025 

870983875 

30.9030743 

9.8476920 

.001047120 

956 

913936 

873722816 

30.9192497 

9.8511280 

.001046025 

957 

915849 

876467493 

30.9354166 

9.8545617 

.001044932 

958 

917764 

879217912 

30.9515751 

9.8579929 

.001043841 

959 

919681 

881974079 

30.9677251 

9.8614218 

.001042753 

960 

921600 

884736000 

30.9838668 

9.8648483 

.001041667 

961 

923521 

887503681 

31.0000000 

9.8682724 

.001040583 

962 

925444 

890277128 

31.0161248 

9.8716941 

.001039501 

963 

927369 

893056347 

31.0322413 

9.  8751135 

.001038422 

964 

929296 

895841344 

31.0483494 

9.8785305 

.001037344 

965 

931225 

898632125 

31.0644491 

9.8819451 

.001036269 

966 

933156 

901428696 

31.0805405 

9.8853574 

.001035197 

967 

935089 

904231063 

31.0966236 

9.8887673 

.001034126 

968 

937024 

907039232 

31.1126984 

9.8921749 

.001033058 

969 

938961 

909853209 

31.1287648 

9.8955801 

.001031992 

970 

940900 

912673000 

31.1448230 

«.  8989830 

.001030928 

971 

942841 

915498611 

31.1608729 

9.9023835 

.001029866 

972 

944784 

918330048 

31.1769145 

9.9057817 

.001028807 

973 

946729 

921167317 

31  .  1929479 

9.9091776 

.001027749 

974 

948676 

924010424 

31.2089731 

9.9125712 

.001026694 

975 

950625 

926859375 

31.2249900 

9.9159624 

.001025641 

976 

952576 

929714176 

31.2409987 

9.9193513 

.001024590 

977 

954529 

932574833 

31.2569992 

9.9227379 

.001023541 

978 

956484 

935441352 

31.2729915 

9.9261222 

.001022495 

979 

958441 

938313739 

31.2889757 

9.9295042 

.001021450 

980 

960400 

941192000 

31.3049517 

9.9328839 

.001020408 

981 

962361 

944076141 

31.3209195 

9.9362613 

.001019368 

982 

964324 

946966168 

31.3368792 

9.9396363 

.001018330 

983 

966289 

949862087 

31.3528308 

9.9430092 

.001017294 

984 

968256 

952763904 

31.3687743 

9.9463797 

.001016260 

985 

970225 

955671625 

31.3847097 

9.9497479 

.001015228 

986 

972196 

958585256 

31.4006369 

9.9531138 

.001014199 

987 

974169 

961504803 

31.4165561 

9.9564775 

.001013171 

988 

976144 

964430272 

31.4324673 

9.9598389 

.001012146 

989 

978121 

967361669 

31.4483704 

9.9631981 

.001011122 

990 

980100 

970299000 

31.4642654 

9.9665549 

.001010101 

991 

982081 

973242271 

31.4801525 

9.9699095 

.001009082 

992 

984064 

976191488 

31.4960315 

9.9732619 

.001008065  ; 

75 


TABLE    VIII. — SQUARES,    CUBES,    ETC. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

993 

986049 

979146657 

31.5119025 

9.9766120 

.001007049 

994 

988036 

982107784 

31.5277655 

9.9799599 

.001006036 

995 

990025 

985074875 

31.5436206 

9.9833055 

.001005025 

996 

992016 

988047936 

31.5594677 

9.9866488 

.001004016 

997 

994009 

991026973 

31.5753068 

9.9899900 

.001003009 

998 

996004 

994011992 

31.5911380 

9.9933289 

.00100.2004 

999 

998001 

997002999 

31  .6069613 

9.9966656 

.001001001 

1000 

1000000 

1000000000 

31.6227766 

10.0000000 

.001000000 

1001 

1002001 

1003003001 

31.6385840 

10.0033322 

.0009990010 

1002 

1004004 

1006012008 

31.6543836 

10.0066622 

.0009980040 

1003 

1006009 

1009027027 

31.6701752 

10.0099899 

.0009970090 

1004 

1008016 

1012J48064 

31.6859590 

10.0133155 

.0009960159 

1005 

1010025 

1015075125 

31.7017349 

10.0166389 

.0009950249 

1006 

1012036 

1018108216 

31.7175030 

10.0199601 

.0009940358 

1007 

1014049 

1021147343 

31.7332633 

10.0232791 

.0009930487 

1008 

1016064 

1024192512    31.7490157 

10.0265958 

.0009920635 

1009 

1018081 

1027243729    31.7647603 

10.0299104 

.0009910803 

1010 

1020100 

1030301COO 

31.7804972 

10.0332228 

.0009900990 

1011 

1022121 

103.3364331    31.7962262 

10.0365330 

.0009891197 

1012 

1024144 

1036433728 

31.8119474 

10.0398410 

.0009881423 

1013 

1026169 

1039509197 

31.8276609 

10.0431469 

.0009871668 

1014 

1028196 

1042590744 

31.8433666 

10.0464506 

.0009861933 

1015 

1030225 

1045678375 

31.8590646 

10.0497521 

.0009852217 

1016 

1032256 

1048772096 

31.8747549 

10.0530514 

.0009842520 

1017 

1034289 

1051871913 

31.8904374 

10.0563485 

.0009832842 

1018 

1036324 

1054977832 

31.9061123 

10.0596435 

.0009823183 

1019 

1038361 

1058089859 

31.9217794 

10.0629364 

.0009813543 

1020 

1040400 

1061208000 

31.9374388 

10.0662271 

.0009803922 

1021 

1042441 

1064&32261 

31.9530906 

10.0695156 

.0009794319 

1022 

1044484 

1067462648 

31.9687347 

10.0728020 

.0009784736 

1023 

1046529 

1070599167 

31.9843712 

10.0760863 

.0009775171 

1024 

1048576 

1073741824 

32.0000000 

10.0793684 

.0009765625 

1025 

1050625 

1076890625 

32.0156212 

10.0826484 

.0009756098 

1026 

1052676 

1080045576 

32.0312348 

10.0859262 

.0009746589 

1027 

1054729 

1083206683 

32.0468407 

10.0892019 

.0009737098 

1028 

1056784 

1086373952 

32.0624391 

10.0924755 

.0009727626 

1029 

1058841 

1089547389 

32.0780298 

10.0957469 

.0009718173 

1030 

1060900 

1092727000 

32.0936131 

10.0990163 

.0009708738 

1031 

1062961 

1095912791 

32.1091887 

10.1022835 

.0009699321 

1032 

1065024 

1099104768 

32.1247568 

10.1055487 

.00  9689922 

1033 

1067089 

1102302937 

32.1403173 

10.1088117 

.0009680542 

1034 

1069156 

1105507304 

32.1558704 

10.1120726 

.0009671180 

1035 

1071225 

1108717875 

32.1714159 

10.1153314 

.0009661836 

1036 

1073296 

1111934656 

32.1869539 

10.1185882 

.0009652510 

1037 

1075369 

1115157653 

32.2024844 

10.1218428 

.0009643202 

1038 

1077444 

1118386872 

32.2180074 

10.1250953 

.0009633911 

1039 

1079521 

1121622319 

32.2335229 

10  .1283457 

.0009624639 

1040 

1081600 

1124864000 

32.2490310 

10.1315941 

.0009615385 

1041 

1083681 

1128111921 

32.2645316 

10.1348403 

.0009606148 

1042 

1085764 

1131366088 

32.2800248 

10.1380845 

.0009596929 

1043 

1087849 

1134626507 

32.2955105 

10.1413266 

.0009587738 

1044 

1089936 

1137893184 

32.3109888 

10.1445667 

.0009578544 

1045 

1092025 

1141166125 

32.3264598 

10.1478047 

.0009569378 

1046 

1094116 

1144445336 

32.3419233 

10.1510406 

.0009560229 

1047 

1096209 

1147730823 

32.3573794 

10.1542744 

.0009551098 

1048 

1098304 

1151022592 

32.3728881 

10.1575062 

.0009541985 

1049 

1100401 

1154320649 

32.3882695 

10.1607359 

.0009532888 

1050 

1102500 

1157625000 

32.4037035 

10.1639636 

.0009523810 

1051 

1104601 

1160935651 

32.4191301 

10.1671893 

.0009514748 

1052 

1106704 

1164252608 

32.4345495 

10.1704129 

.0009505703 

1053 

1108809 

1167575877 

32.4499615 

10.1736344 

.0009496676 

1054 

1110916 

1170905464 

32.4653662 

10.1768539 

.0009487666 

76 


TABLE    IX. — LOGARITHMS    OF    NUMBERS. 


No. 

100  L.  000.] 

.No.  109  L.  040. 

N. 

0 

1 

2 

8         4 

5 

6 

7 

8 

9 

Diff. 

100 

000000 

0434 

0868 

1301     1734 

2166 

2598 

3029 

3461 

3891 

432 

1 

4321 

4751 

5181 

5609     6038 

6466 

6894 

7321 

7748 

8174 

426 

2 

8600 

9026 

9451 

9876 

0300 

0724 

1147 

1570 

1993 

2415 

AS>A 

3 

012837 

3259 

3680 

4100     4521 

4940 

5360 

5779 

6197 

6616 

420 

4 

7033 

7451 

7868 

8284     8700 

9116 

9532 

9947 

0361 

0775 

416 

5 

021189 

1603 

2016 

2428     2841 

3252 

3664 

4075 

4486 

4896 

412 

6 

5306 

5715 

6125 

6533     6942 

7350 

7757 

8164 

8571 

8978 

408 

9384 

9789 

0195 

0600     1004 

1408 

1812 

2216 

261S 

3021 

404 

8 

033424 

3826 

4227 

4628     5029 

5430 

5830 

6230 

6629 

7028 

400 

g 

7426 

7825 

8223 

8620     9017 

9414 

9811 

04 

0207 

0602 

0998 

397 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

434 

43.4 

86.8 

130.2 

173.6 

217.0 

260.4 

3( 

B.8 

347.2 

390.6 

433 

43.3 

86.6 

12 

9.9 

173.2 

216.5 

259 

8 

3( 

B.I 

346.4 

389.7 

432 

43.2 

86.4 

12 

9.6 

172.8 

216.0 

259 

2 

3( 

)2.4 

345.6 

388.8 

431 

43.1 

86.2 

129.3 

172.4 

215.5 

258 

0 

301.7 

344.8 

387.9 

430 

43.0 

86.0 

12 

9.0 

172.0 

215.0 

258 

0 

3( 

)1.0 

344.0 

387.0 

429 

42.9 

85.8 

128  7 

171.6 

214.5 

257 

4 

300.3 

343.2 

386.1 

428 

42.8 

85.6 

12 

8.4 

171.2 

214.0 

256 

8 

21 

)9.6 

342.4 

385.2 

427 

42.7 

85.4 

128.1 

170.8 

213.5 

256.2 

298.9 

341.6 

384.3 

426 

42.6 

85.2 

12 

7.8 

170.4 

213.0 

255 

6 

2 

38.2 

340.8 

383.4 

425 

42.5 

85.0 

127.5 

170.0 

212.5 

255 

0 

297.5 

340.0 

382.5 

424 

42.4 

84.8 

127.2 

169.6 

212.0 

254 

4 

296.8 

339.2 

381.6 

423 

42.3 

84.6 

12 

6.9 

169.2 

211.5 

253 

8 

2< 

36.1 

338.4 

380:7 

422 

42.2 

84.4 

126.6 

168.8 

211.0 

253.2 

295.4 

337.6 

379.8 

421 

42.1 

84.2 

12 

6.3 

168.4 

210.5 

252 

6 

2< 

34.7 

336.8 

378.9 

420 

42.0 

84.0 

126.0 

168.0 

210.0 

252.0 

294.0 

336.0 

378.0 

419 

41.9 

83.8 

12 

5.7 

167.6 

209.5 

251 

4 

2< 

33.3 

335.2 

377.1 

418 

41.8 

83.6 

12 

5.4 

167.2 

209.0 

250 

8 

2 

32.6 

334.4 

376.2 

417 

41.7 

as.  4 

125.1 

166.8 

208.5 

250 

2 

291.9 

333.6 

375.3 

416 

41.6 

83.2 

12 

4.8 

166.4 

208.0 

249 

6 

2< 

31.2 

332.8 

374.4 

415 

41.5 

83.0 

124.5 

166.0 

207.5 

249 

0 

290.5 

332.0 

373.5 

414 

41.4 

82.8 

124.2 

165.6 

207.0 

248 

4 

21 

39.8 

asi.2 

372.6 

413 

41.3 

82.6 

12 

3.9 

165.2 

206.5 

247 

8 

2 

39.1 

330.4 

371.7 

412 

41.2 

82.4 

12 

3.6 

164.8 

206.0 

247 

2 

2 

38.4 

329.6 

370.8 

411 

41.1 

82.2 

123.3 

164.4 

205.5 

246 

6 

2 

37.7 

328.8 

369.9 

410 

41.0 

82.0 

12 

3.0 

164.0 

205.0 

246 

.0 

a 

37.0 

328.0 

369.0 

409       40.9 

81.8 

122.7 

163.6 

204.5 

245 

.4 

286.3 

327.2 

368.1 

408 

40.8       81.6 

12 

•2.4 

163.2       204.0 

244 

.8 

2 

35.6 

326.4 

367.2 

407 

40.7  I    81.4 

12 

•2.1 

162.8 

203.5 

244 

.2 

2 

34.9 

325.6 

366.3 

406 

40.6 

81.2 

121.8 

162.4 

203.0 

243 

6 

284.2 

324.8 

365.4 

405 

40.5 

81.0 

121.5 

162.0 

202.5 

243.0 

283.5 

324.0 

364.5 

404 

40.4 

80:8 

121.2 

161.6 

202.0 

242 

.4 

282.8 

323.2 

363.6 

40t 

5 

40.3 

80  6 

IS 

50.9 

161.2 

201.5 

241 

.8 

2 

82.1 

322.4 

362.7 

40$ 

! 

40.2 

80.4 

IS 

JO.  6 

160.8 

201.0 

241 

2 

2 

81.4 

321.6 

361.8 

401 

40.1 

80.2 

120.3 

160.4 

200.5 

240.6 

280.7 

320.8 

360.9 

4(K 

) 

40.0 

80-0 

is 

JO.O 

160.0 

200.0 

240 

.0 

2 

80.0 

320.0     360.0 

39< 

) 

39.9  \     79.8 

1] 

L9.7 

159.6 

199.5 

239 

.4 

2 

79.3 

319.2  !  359.1 

39! 

J 

39  8  '     79.6 

119.4 

159.2 

199.0 

238 

.8 

278.6 

318.4  !  358.2 

39 

39.7       79.4 

1 

9.1 

158.8 

198.5 

238 

.2       2 

77.9 

317.6     357.3 

39 

3       3!).  (i       79.2 

1 

18.8 

158.4 

198.0 

237 

.6 

2 

77.2 

316.8  i  356.4 

395       39.5       79.0         118.5 

158.0       197.5       237.0       276.5       316  0  !  355.5 

TABLE    IX. — LOGARITHMS    OF   NUMBERS. 


No. 

110  L.  041.] 

[No.  119  L.  078. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

110 

041393 

1787 

2182 

2576 

2969 

3362     3755 

4148 

4540 

4932 

393 

1 

5323 

5714 

6105 

6495 

6885 

7275 

7664 

8053 

8442 

8830 

390 

2 

9218 

9606 

9993 

038C 

0766 

1153     i^sa 

1924 

2309 

2694 

386 

3 

053078 

3463 

3846 

4230 

4613 

4996 

5378 

5760 

6142 

6524 

383 

4 

6905 

7286 

7666 

8046 

8426 

8805 

9185 

9563 

9942 

0320 

379 

5 

060698 

1075 

1452 

1829 

2206 

2582 

2958 

3333 

3709 

4083 

376 

6 

4458 

4832 

5206 

5580 

5953 

6326 

6699 

7071 

7443 

7815 

373 

7 

8186 

8557 

8928 

9298 

9668 

0038 

0407 

0776 

1145 

1514 

370 

8 

071882 

2250 

2617 

2985 

3352 

3718 

4085 

4451 

4816 

5182 

366 

9 

5547 

5912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

363 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

395 
394 

39.5 
39.4 

79.0 

78.8 

118.5 
118.2 

158.0 
157.6 

197.5 
197.0 

237.0 
236.4 

276.5 

275.8 

316.0 
315.2 

355.5 
354.6 

393 

39.3 

78.6 

11 

7.9 

157.2 

196.5 

235 

.8 

2 

75.1 

314.4 

353.7 

392 

39.2 

78.4 

117.6 

156.8 

196.0 

235 

.2 

274  .4 

313.6 

352.8 

391 

39.1 

78.2 

11 

7.3 

156.4 

195.5 

234 

.6 

2 

73.7 

312.8 

351.9 

390 

39.0 

78.0 

11 

7.0 

156.0 

195.0 

234 

.0 

273.0 

312.0 

a5i.o 

389 

38.9 

77.8 

11 

6.7 

155.6 

194.5 

233 

.4 

2 

72.3 

311.2 

350.1 

388 

38.8 

77.6 

11 

6.4 

155.2 

194.0 

232 

.8 

2 

71.6 

310.4 

349.2 

387 

38.7 

77.4 

116.1 

154.8 

193.5 

232.2 

270.9 

309.6 

348.3 

386 

38.6 

77.2 

11 

5.8 

154.4 

193.0 

231 

.6 

2 

70.2 

308.8 

347.4 

38E 

38.5 

77.0 

115.5 

154.0 

192.5 

231 

.0 

269.5 

308.0 

346.5 

384 

38.4 

76.  £ 

! 

115.2 

153.6 

192.0 

230.4 

268.8 

307.2 

345.6 

38? 

38.3 

76.  ( 

) 

11 

4.9 

153.2 

191.5 

228 

.8 

2 

68.1 

306.4 

344.7 

382 

38.2 

76.4 

[ 

1] 

4.6 

152.8 

191.0 

22S 

.2 

2 

67.4 

305.6     343.8 

381 

38.1 

76.2 

114.3 

152.4 

190.5 

228.6 

266.7 

304.8  !  342.9 

38C 

) 

38.0 

76.  ( 

) 

1] 

4.0 

152.0 

190.0 

228 

.0 

2 

66.0 

304.0 

342.0 

379 

37.9 

75.* 

] 

113.7 

151.6 

189.5 

227.4 

265.3 

303.2 

341.1 

37* 

5 

37.8 

75.  ( 

> 

1] 

3.4 

151.2 

189.0 

226 

.8 

2 

64.6 

302.4 

340.2 

gjr 

37.7 

75.' 

t 

1] 

3.1 

150.8 

188.5 

226 

.2 

2 

63.9 

301.6 

339.3 

5 

37.6 

75.  J 

I 

112.8 

150.4 

188.0 

225.6 

263.2 

300.8 

338.4 

375 

37.5 

75.0 

112.5 

150.0 

187.5 

225.0 

262.5 

300.0 

337.5 

374 

37.4 

74.* 

\ 

112.2 

149.6 

187.0 

224.4 

261.8 

299.2 

336.6 

37; 

5 

37.3 

74.  ( 

1] 

LI.  9 

149.2 

186.5 

22? 

.8 

2 

61.1 

298.4 

335.7 

372 

37.2 

74.4 

111.6 

148.8 

186.0 

223.2 

260.4 

297.6 

334.8 

37] 

L 

37.1 

74.5 

> 

1 

LI.  3 

148.4 

185.5 

222 

.6 

2 

59.7 

296.8 

a33.9 

370 

37.0 

74.0 

111.0 

148.0 

185.0 

222.0 

259.0 

296.0 

333.0 

36< 

) 

36.9 

73.* 

\ 

1 

10.7 

147.6 

184.5 

22] 

.4 

2 

58.3 

295.2 

332.1 

3$ 

j 

36.8 

73.  ( 

5 

1 

0.4 

147.2 

184.0 

226 

2 

57.6 

294.4 

331.2 

S67 

36.7 

73.4 

110.1 

146.8 

183.5 

220.2 

256.9 

293.6 

330.3 

36 

36.6 

73.  J 

I 

1( 

)9.8 

146.4 

183.0 

21t 

.6 

2 

56.2 

292.8 

329.4 

SG5 

36.5 

73.0 

109.5 

146.0 

182.5 

219.0 

255.7 

292.0 

328.5 

364 

36.4 

72. 

5 

109.2 

145.6 

182.0 

218.4 

254.8 

291.2 

327.6 

36 

3 

36.3 

72. 

3 

1( 

)8.9 

145.2 

181.5 

217 

.8 

2 

54.1 

290.4 

326.7 

362 

36.2 

72.4 

108.6 

144.8 

181.0 

217 

.2 

253.4 

289.6 

325.8 

36 

I 

36.1 

72. 

2 

1( 

)8.3 

144.4 

180.5 

216 

.6 

2 

52.7 

288.8 

324.9 

36 

) 

36.0 

72. 

) 

H 

)8.0 

144.0 

180.0 

216 

.0 

2 

52.0 

288.0 

324.0 

359 

35.9 

71. 

3 

107.7 

143.6 

179.5 

215.4 

251.3 

287.2 

«23.1 

351 

3 

35.8 

71. 

5 

1 

37.4 

143.2 

179.0 

214 

.8 

2 

50.6 

286.4 

322.2 

as 

7 

35.7 

71. 

4 

1 

37.1 

1428       178.5 

214 

.2 

2 

49.9 

285.6 

321.3 

356 

35.6 

71.2 

106.8 

142.4 

178.0 

213.6 

249.2 

284.8 

320.4 

TABLE    IX. — LOGARITHMS    OF    NUMBERS. 


No.  120  L.  079.] 

[No.  134  L.  130. 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Diff. 

1 

120 

079181 

9543 

9904 

0266 

0626 

I    0987 

1347 

1707 

2067 

2426 

360 

1 

082785 

3144 

3503 

3861 

4219 

1    4576 

4934 

5291 

5647 

6004 

357 

2 

6360 
9905 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

355 

0258 

0611 

0963 

1315 

1    1667 

2018 

2370 

2721 

3071 

352 

4 

093422 

3772 

4122 

4471 

4820 

5169 

5518 

5806 

6215 

6562 

349 

5 

6910 

7257 

7004 

7951 

8298 

8644 

8990 

9335 

9681 

0026 

346 

6 

100371 

0715 

1059 

1403 

1747 

2091 

2434 

2777 

3119 

3462 

343 

7 

3804 

4146 

4487 

4828 

5169 

i  5510 

5851 

6191 

6531 

6871 

341 

8 

7210 

7549 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

0253 

338 

9 

110590 

0926 

1363 

1599 

1934 

2270 

2605 

2940 

3275 

3609 

335 

130 

3943 

4277 

4611 

4944 

5278 

5611 

5943 

6276 

6608 

6940 

333 

7271 

7603 

7934 

8265 

8595 

1  8926 

9256 

3586 

9915 

0245 

330 

2 

120574 

0903 

1231 

1560 

1888 

2216 

2544 

2871 

3198 

3525 

328 

3 

3852 

4178 

4504 

4830 

5156 

5481 

5806 

6131 

6456 

6781 

325 

4 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

9090 

13 

0012 

323 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

355 

35.5 

71.0 

106.5 

142.0 

177.5 

213.0 

248.5 

284.0 

319.5 

354 

35.4 

70.8 

106 

.2 

141.6 

177.0 

212.4 

247.8 

283.2 

318.6 

353 

35.3 

70.6 

105 

.9 

141.2 

176.5 

211.8 

247.1 

282.4 

317.7 

352 

35.2 

70.4 

105.6 

140.8 

176.0 

211.2 

246.4 

281.6 

316.8 

351 

35.1 

70.2 

105 

.3 

140.4 

175.5 

210.6 

245.7 

280.8 

315.9 

350 

35.0 

70.0 

105.0 

140.0 

175.0 

210.0 

245.0 

280.0 

315.0 

349 

34.9 

69.8 

104 

.7 

139.6 

174.5 

209.4 

244.3 

279.2 

314.1 

348 

34.8 

69.6 

104.4 

139.2 

174.0 

208.8 

243.6 

278.4 

313.2 

347 

34.7 

69.4 

104 

.1 

138.8 

173.5 

208.2 

242.9 

277.6 

312.3 

346 

34.6 

69.2 

103.8 

138.4 

173.0 

207.6 

242.2 

276.8 

311.4 

345 

34.5 

69.0 

103.5 

138.0 

172.5 

207.0 

241.5 

276.0 

310.5 

344 

34.4 

68.8 

103.2 

137.6 

172.0 

206.4 

240.8 

275.2 

309.6 

343 

34.3 

68.6 

102 

.9 

137.2 

171.5 

205.8 

240.1 

274.4 

308.7 

342 

34.2 

68.4 

102.6 

136.8 

171.0 

205.2 

239.4 

273.6 

307.8 

341 

34.1 

68.2 

102 

.8 

136.4 

170.5 

204.6 

238.7 

272.8 

306.9 

340 

34.0 

68.0 

102.0 

136.0 

170.0 

204.0 

238.0 

272.0 

306.0 

339 

33.9 

67.8 

101 

.7 

135.6 

169.5 

203.4 

237.3 

271.2 

305.1 

338 

33.8 

67.6 

101 

.4 

135.2 

169.0 

202.8 

236.6 

270.4 

304.2 

337 

33.7 

67.4 

101 

.1 

134.8 

168.5 

202.2 

235.9 

•  269.6 

303.3 

336 

33.6 

67.2 

100.8 

134.4 

168.0 

201.6 

235.2 

268.8 

302.4 

335 

33.5 

67.0 

100.5 

134.0 

167.5 

201.0 

234.5 

268.0 

301.5 

334 

33.4 

66.8 

100 

.2 

133.6 

167.0 

200.4 

233.8 

267.2 

300.6 

333 

33.3 

66.6 

99 

.9 

133.2 

166.5 

199.8 

233.1 

266.4 

299.7 

332 

33.2 

66.4 

99.6 

132.8 

166.0 

199.2 

232.4 

265.6 

298.8 

331 

33.1 

66.2 

99 

.3 

132.4 

165.5 

198.6 

231.7 

264.8 

297.9 

330 

33.0 

66.0 

99.0 

132.0 

165.0 

198.0 

231.0 

264.0 

297.0 

329 

32.9 

65.8 

98 

.7 

131.6 

164.5 

197.4 

230.3 

263.2 

296.1 

328 

32.8 

65.6 

98 

.4 

131.2 

164.0 

196.8 

229.6 

262.4 

295.2 

327 

32.7 

65.4 

98.1 

130.8 

163.5       196.2 

228.9 

361.6 

294.3 

326 

32.6 

65.2 

97.8 

130.4 

103.0       195.6 

228.2 

260.8 

293.4 

325 

32.5 

65.0 

97.5 

130.0 

162.5       195.0 

227.5 

260.0 

292.5 

324 

32.4 

64.8 

97 

.2 

129.6 

162.0  i     194.4 

226.8 

259.2 

291.6 

323 

32.3 

64.6 

96 

.9 

129.2 

161.5        193.8 

226.1 

<• 

258.4 

290.7 

322 

32.2 

64.4 

96.6 

128.8       161.0 

193.2 

225.4 

257.6 

289.8 

TABLE    IX. — LOGARITHMS    OF    NUMBERS. 


No.  135  L.  130.] 

[No.  149  L.  175. 

N. 

0 

1          2 

3 

4 

5J     . 

7 

8 

9 

Diff. 

135 

130aS4 

0655     0977 

1298 

1619 

1939  i  2260 

2580 

2900 

3219 

321 

6 

3539 

3858     4177 

4496 

4814 

5133  i  5451 

5769 

6086 

6403 

318 

7 

6721 

7037     7354 

7671 

7987 

8303     8618 

8934 

9249 

9564 

316 

1  

0194     0508 

0822 

1136 

1450     1763 

2076 

2389 

2702 

314 

9 

143015 

3327     3639 

3951 

4263 

4574     4885 

5196 

5507 

5818 

311 

140 

6128 
9219 

6438     6748 
9527     9835 

7058 

7367 

7676     7985 

8294 

8603 

8911 

309 

0142 

0449 

0756     1fvi3 

1370 

1676 

1982 

307 

2  1  152288 

2594     2900 

3205 

3510 

3815 

4120 

4424 

4728 

5032 

305 

3         5336 

5640     5943 

6246 

6549 

6852 

7154 

7457 

7759 

8061 

303 

4 

8362 

8664     8965 

9266 

9567 

9868 

0168 

0469 

0769 

1068 

301 

5 

161368 

1667     1967 

2266 

2564 

2863 

3161 

3460 

3758 

4055 

299 

6 

4353 

4650     4947 

5244 

5541 

5838 

6134 

6430 

6726 

7022 

297 

7 

7317 

7613     7908 

8203 

8497 

8792 

9086 

9380 

9674 

9968 

295 

8 

170262 

0555      0848 

1141 

1434 

1726 

2019 

2311 

2603 

2895 

293 

9 

3186 

3478     3769 

4060 

4351 

4641 

4932 

5222 

5512 

5802 

291 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

321 

32.1 

64.2 

96.3 

128.4 

160.5 

192.8 

224.7 

256.8 

288.9 

320 

32.0 

64.0 

96.0 

128.0 

160.0 

192.0 

224.0 

256.0 

288.0 

319 

31.9 

63.8 

95.7 

127.6 

159.5 

191.4 

2S 

8.3 

255.2 

287.1 

318 

31.8 

63.6 

95.4 

127.2 

159.0 

190.8 

2$ 

52.6 

254.4 

286.2 

317 

31.7 

63.4 

95.1 

126.8 

158.5 

190.2 

221.9 

253.6 

285.3 

316 

31.6 

63.2 

94.8 

126.4 

158.0 

189.6 

2$ 

>1.2 

252.8 

284.4 

315 

31.5 

63.0 

94.5 

126.0 

157.5 

189.0 

220.5 

252.0 

283.5 

314 

31.4 

62.8 

94.2 

125.6 

157.0 

188.4 

21 

9.8 

251.2 

282.6 

318 

31.3 

62.6 

93.9 

125.2 

156.5 

187.8 

21 

9.1 

250.4 

281.7 

312 

31.2 

62.4 

93.6 

124.8 

156.0 

187.2 

218.4 

249.6 

280.8 

311 

31.1 

62.2 

93.3 

124.4 

155.5 

186.6 

217.7 

248.8 

279.9 

310 

31.0 

62.0 

93.0 

124.0 

155.0 

186.0 

217.0 

248.0 

279.0 

309 

30.9 

61.8 

92.7 

123.6 

154.5 

185.4 

21 

6.3 

247.2 

278.1 

308 

30.8 

61.6 

92.4 

123.2 

154.0 

184.8 

215.6 

246.4 

277.2 

307 

30.7 

61.4 

92.1 

1S2.8 

153.5 

184.2 

21 

4.9 

245.6 

276.3 

306 

30.6 

61.2 

91.8 

122.4 

153.0 

183.6 

214.2 

244.8 

275.4 

305 

30.5 

61.0 

91.5 

122.0 

152.5 

183.0 

21 

3.5 

244.0 

274,5 

304 

30.4 

60.8 

91.2 

121.6 

152.0 

182.4 

21 

2.8 

243.2 

273.6 

303 

30.3 

60.6 

90.9 

121.2 

151.5 

181.8 

212.1 

242.4 

272.7 

302 

30.2 

60.4 

90.6 

120.8 

151.0 

181.2 

211.4 

241.6 

271.8 

301 

30.1 

60.2 

90.3 

120.4 

150.5 

180.6 

210.7 

240.8 

270.9 

300 

30.0       60.0 

90.0 

120.0 

150.0 

180.0 

210.0 

240.0 

270.0 

299 

29.9       59.8 

89.7 

119.6 

149.5 

179.4 

2( 

)9.3 

239.2 

269.1 

298 

29.8 

59.6 

89.4 

119.2 

149.0 

178.8 

2( 

».6 

238.4 

268.2 

297 

29.7 

59.4 

89.1 

118.8 

148.5 

178.2 

207.9 

237.6 

267.3 

296 

29.6       59.2 

88.8 

118.4 

148.0 

177.6 

2( 

)7.2 

236.8 

266.4 

295 

29.5  i     59.0 

88.5 

118.0 

147.5 

177.0 

206.5 

236.0 

265.5 

294 

29.4       58.8 

88.2 

117.6 

147.0 

176.4 

if 

)5.8 

235.2 

264.6 

293 

29.3 

58.6 

87.9 

117.2 

146.5 

175.8 

i 

)5.1 

234.4 

263.7 

292 

29.2 

58.4 

87.6 

116.8 

146.0 

175.2 

204.4 

233.6 

262.8 

291 

29.1 

58.2 

87.3 

116.4 

145.5 

174.6 

203.7 

232.8 

261.9 

290 

29.0 

58.0        87.0 

116.0 

145.0 

174.0 

9 

)3.0 

232.0 

261.0 

289 

28.9 

57.8 

86.7 

115.6 

144.5 

173.4 

2( 

)2.3 

231.2 

260.1 

288 

28.8 

57.6         86.4 

115.2 

144.0 

172.8 

201.6 

230.4 

259.2 

287 

28.7 

57.4 

86.1 

114.8 

143.5 

172.2 

2( 

X).9 

229.6 

258.3 

286 

28.6 

57.2         85.8 

114.4       143.0       171.6 

200.2  j    228.8 

257.4 

TABLE   IX.  — LOGARITHMS   OF   NUMBERS. 


No.  150  L.  176.] 

[No.  169  L.  230. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

150~ 

176091 

6381  6670 

6959 

7248 

7536 

7825 

8113 

8401 

8689 

289 

8977 

9264 

OfW>   QW'-IQ 

0126  1  0413 

0699 

0986 

1272 

•JRKQ 

907 

2 

181844 

2129  2415 

2700 

2985   3270 

3555 

3839 

4123 

lOOD 

4407 

*o< 
285 

3 

4691 

4975 

5259 

5542 

5825 

6108 

6391 

6674 

6956 

7239 

283 

4 

7521 

7803 

8084 

8647 

8928 

9209 

9490 

9771 

0051 

281 

5 

190332 

0612 

0892 

1171 

1451 

1730 

2010 

2289 

2567 

2846 

279 

6 

3125 

3403 

3681 

3959 

4237 

4514 

4792 

5069 

5346 

5623 

378 

7 
g 

5900 
8657 

6176 
8932 

6453 
9206 

6729 

9481 

7005 
9755 

j  7281 

7556 

7832 

8107 

8382 

276 

0029 

0303 

0577 

0850 

1124 

274 

9 

201397 

1670 

1943 

2216 

2488 

2761 

3033 

3305 

3577 

3848 

272 

160 

4120 

4391 

4663 

4934 

5204 

5475 

5746 

6016 

6286 

6556 

271 

1 
2 

6826 
9515 

7096 
9783 

7365 

7634 

7904 

8173 

8441 

8710 

8979 

9247 

269 

i  0051 

0319 

0586 

0853 

1121 

1388 

1654 

1921 

267 

3 

212188 

2454  2720 

2986 

3252 

3518 

3783 

4049 

4314 

4579 

266 

4 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

264 

5 

7484 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9846 

262 

6 

220108 

0370 

0631 

0892 

1153 

1414 

1675 

1936 

2196 

2456 

261 

7 

2716 

2976 

3236 

3496 

3755 

4015 

4274 

4533 

4792 

5051 

259 

8 

5309 

5568 

5826 

6084- 

6342 

6600 

6858 

7115 

7372 

7630 

258 

9 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 



23 

0193 

256 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

285 

28.5 

57.0 

85.5 

114.0 

142.5 

171.0 

199.5 

228.0 

256.5 

284 

28.4 

56.8 

85.2 

113.6 

142.0 

170.4 

198.8 

227.2 

255.6 

283 

28.3 

56.6 

84.9 

113.2 

141.5 

169.8 

198.1 

226.4 

254.7 

282 

28.2 

56.4 

84.6 

112.8 

141.0 

169.2 

197.4 

225.6 

253.8 

281 

28.1 

56.2 

84.3 

112  4 

140.5 

168.6 

196.7 

224.8 

252.9 

280 

28.0 

56.0 

84.0 

112.0 

140.0 

168.0 

196.0 

224.0 

252.0 

279 

27.9 

55.8 

83.7 

111.6 

139.5 

167.4 

195.3 

223.2 

251.1 

278 

27.8 

55.6 

83.4 

111.2 

139.0 

166.8 

194.6 

222.4 

250.2 

277 

27.7 

55.4 

83.1 

110.8 

138.5 

166.2 

193.9 

221.6 

249.3 

276 

27.6 

55.2 

82.8 

110.4 

138.0- 

165.6 

193.2 

220.8 

248.4 

275 

27.5 

55.0 

82.5 

110.0 

137.5 

165.0 

192.5 

220.0 

247.5 

274 

27.4 

54.8 

82.2 

109.6 

137.0 

164.4 

191.8 

219.2 

246.6 

273 

27.3 

54.6 

81.9 

109.2 

136.5 

163.8 

191.1 

218.4 

245.7 

272 

27.2 

54.4 

81.6 

108.8 

136.0 

163.2 

190.4 

217.6 

244.8 

271 

27.1 

54.2 

81.3 

108.4 

135.5 

162.6 

189.7 

216.8 

243.9 

270 

27.0 

54.0 

81.0 

108.0 

135.0 

162.0 

189.0 

216.0 

243.0 

269 

26.9 

53.8 

80.7 

107.6 

134.5 

161.4 

188.3 

215.2 

242.1 

268 

26.8 

53.6 

80.4 

107.2 

134.0 

160.8 

187.6 

214.4 

241.2 

267 

26.7 

53.4 

80.1 

106.8 

133.5 

160.2 

186.9 

213.6 

240.3 

266 

26.6 

53.2 

79.8 

106.4 

133.0 

159.6 

186.2 

212.8 

239.4 

265 

26.5 

53.0 

79.5 

106.0 

132.5 

159.0 

185.5 

212.0 

238.5 

264 

26.4 

52.8 

79.2 

105.6 

132.0 

158.4 

184.8 

211.2 

237.6 

263 

26.3 

52.6 

78.9 

105.2 

131.5 

157.8 

184.1 

210.4 

236.7 

262 

26.2 

52.4 

78.6 

104.8 

131.0 

157.2 

183.4 

209.6 

235.8 

261 

26.1 

52.2 

78.3 

104.4 

130.5 

156.6 

182.7 

208.8 

234.9 

260 

26.0 

52.0 

78.0 

104.0 

130.0 

156.0 

182.0 

208.0 

234.0 

259 

25.9 

51.8 

77.7 

103.6 

129.5 

155.4 

181.3 

207.2 

233.1 

258 

25.8 

51.6 

77.4 

103.2 

129.0 

154.8 

180.6 

206.4 

232.2 

257 

25.7 

51.4 

102.8 

128.5 

154.2 

179.9 

205.6 

231.3 

256 

25.6 

51.2 

76i8 

102.4 

128.0   153.6   179.2  !  204.8 

230.4 

255 

25.5 

51.0 

76.5 

102.0 

1£7.5  j  153.0   178.5  |  204.0 

229.5 

81 


TABLE    IX. — LOGARITHMS    OF    NUMBERS. 


No.  170  L.  230.]                                   [No.  189  L.  278. 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Diff. 

170 

230449 

0704 

0960 

1215 

1470 

1724 

1979 

2234 

2488 

2742   255 

1 

2996 

3250 

3504 

3757 

4011 

4264 

4517 

4770 

5023 

5276 

253 

2 

5528 

5781 

6033 

6285 

6537 

6789 

7041 

7292 

7544 

7795 

252 

3 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

0050 

0300 

OKA 

4 

240549 

0799 

1048 

1297 

1546 

1795 

2044 

2293 

2541 

2790 

(COU 

249 

5 

3038 

3286 

35J34 

3782 

4030 

4277 

4525 

4772 

5019 

5266 

248 

G 

5513 

5759 

G006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

246 

7 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

0176 

OAK 

8 

250420 

0664 

0908 

1151 

1395 

1638 

1881 

2125 

2368 

2610 

/*4O 

243 

9 

2863 

3096 

3338 

3580 

3822 

4064 

4306 

4548 

4790 

5031 

242 

180 

5273 

5514 

5755 

5996 

6237 

6477 

6718 

6958 

7198 

7439 

241 

1 

7679 

7918 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239 

2 

260071 

0310 

0548 

~0787~ 

1025 

1263 

1501 

1739 

1976 

2214 

238  I 

3 

2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

237 

4 

4818 

5054 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235 

5 

7172 

7406 

7641 

7875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

G 

9513 

9746 

9980 

0213 

0446 

0679 

0912 

1~ldA 

1^77 

ifino 

QQO 

7 

271842 

2074 

2306 

2538 

2770 

3001 

3233 

J.14<* 

3464 

lo(  i 

3696 

jouy 
3927 

<*OO 

232 

8 

4158 

4389 

4620 

4850 

5081 

5311 

5542 

5772 

6002 

6232 

230 

9 

6462 

6692 

6921 

7151 

7380 

7609 

7838 

8067 

8296 

R525 

229 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

255 
254 

25.5 
25.4 

51.0 

50.8 

76.5 
76.2 

102.0 
101.6 

127.5 
127.0 

153.0 
152.4 

178.5 

177.8 

253 

25.3 

50.6 

75.9 

101.2 

126.5 

151.8 

177.1 

202  .'4 

227!  7 

252 

25.2 

50.4 

75.6 

100.8 

126.0 

151.2 

176.4 

201.6 

226.8 

251 

25.1 

50.2 

75.3 

100.4 

125.5 

150.6 

175.7 

200.8 

225.9 

250 

25  0 

50.0 

75.0 

100.0 

125.0 

150.0 

175.0 

200.0 

225.0 

249 

24.9 

49.8 

74.7 

99.6 

124.5 

149.4 

174.3 

199.2 

224.1 

248 

24.8 

49.6 

74.4 

99.2 

124.0 

148.8 

173.6 

198.4 

223.2  ; 

247 

24.7 

49.4 

74.1 

98.8 

123.5 

148.2 

172.9 

197.6 

222.3 

246 

24.6 

49.2 

73.8 

98.4 

123.0 

147.6 

172.2 

196.8 

221.4 

245 

24.5 

49.0 

73.5 

98.0 

122.5 

147.0 

171.5 

196.0 

220.5 

244 

24.4 

48.8 

73.2 

97.6 

122.0 

146.4 

170.8 

195.2 

219.6 

243 

24.3 

48.6 

72.9 

97.2 

121.5 

145.8 

170.1 

194.4 

218.7  . 

242 

24.2 

48.4 

72.6 

96.8 

121.0 

145.2 

169.4 

193.6 

217.8 

241 

24.1 

48.2 

72.3 

96.4 

120.5 

144.6 

168.7 

192.8 

216.9 

240 

24.0 

48.0 

72.0 

96.0 

120.0 

144.0 

168.0 

192.0 

216.0 

239 

23.9 

47.8 

71.7 

95.6 

119.5 

143.4 

167.3 

191.2 

215.1 

2:38 

23.8 

47.6 

71.4 

95.2 

119.0 

142.8 

166.6 

190.4 

214.2 

237 

23.7 

47.4 

71.1 

94.8 

118.5 

142.2 

165.9 

189.6 

213.3 

236 

23.6 

47.2 

70.8 

94.4 

118.0 

141.6 

165.2 

188.8 

212.4 

235 

23.5 

47.0 

70.5 

94.0 

117.5 

141.0 

164.5 

188.0 

211.5 

234 

23.4 

46.8 

70.2 

93.6 

117.0 

140.4 

163.8 

187.2 

210.6 

2-33 

23.3 

46.6 

69.9 

93.2 

116.5 

139.8 

163.1 

186.4 

209.7 

232 

23.2 

46.4 

69.6 

92.8 

116.0 

139.2 

162.4 

185.6 

208.8 

231 

23.1 

46.2 

69.3 

92.4 

115.5 

138.6 

161.7 

184.8 

207.9 

230 

23.0 

46.0 

69.0 

92.0 

115.0 

138.0 

161.0 

184.0 

207.0 

229 

22.9 

45.8 

68.7 

91.6 

114.5 

137.4 

160.3 

183.2 

206.1 

228 

22.8 

45.6 

68.4 

91.2   114.0 

136.8 

159.6 

182.4 

205.2 

227 

22.7 

45.4 

68.1 

90.8 

113.5 

136.2 

158.9 

181.6 

204.3 

226 

22.6 

45.2 

67.8 

90.4 

113.0 

135.6 

158  2 

180.8 

203.4 

TABLE    IX. — LOGARITHMS  -OF    NUMBERS. 


No.  190  L.  278.]                                                                                   [No.  214  L.  332. 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Diff. 

190 

278754 

8982 

9211 

9439 

9667 

!  9895 

0351 

0578 

0806 

228 

1 

281033 

1261 

1488 

1715 

1942 

2169 

2849 

3075 

227 

2 

3301 

3527 

3753 

§979 

4205 

4431 

4056 

4882 

5107 

5332 

226 

3 

5557 

5782 

6007 

6232 

0450       6081 

6906 

7130 

7354 

7578 

225 

4 

7802 

8026 

8249 

8473 

8090  !    8920 

9143 

9306 

9589 

9812 

223 

5 

290035 

0257 

0480 

0702 

0925 

1147 

1369 

1591 

1813 

2034 

222 

6 

2256 

2478 

2699 

2920 

3141 

3303 

8584 

3804 

4025 

4246 

221 

4466 

4687 

4907 

5127 

5:347 

i  5567 

5787 

6007 

6226 

6446 

220 

8 

6665 

OQK.Q 

6884 
on**"! 

7104 
9289 

7323 
9507 

7542 
9725 

7761 
9943 

7979 

8198 

8416 

8635 

219 

ooDO 

JU«  1 

0161 

0378 

0595 

0813 

218 

200 

301030 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

217 

1 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

216 

2 

5351 

5566 

5781 

5996 

6211 

(5425      6039 

6854 

7068 

7282 

215 

3 

7496 

QfiQA 

7710 

7924 

8137 

8351 

|  8564 

8778 

8991 

9204 

9417 

213 

yOoU 

9843 

0056 

0268 

04S1 

0693 

0906 

1118 

1330 

1542 

212 

5 

311754 

1966 

2177 

2389 

2000 

2812 

3023 

3234 

3445 

3656 

211 

6 

3867 

4078 

4239 

4499 

4710 

4920 

5130 

5340 

5551 

5760 

210 

5970 

6180 

6390 

6599 

6809 

•  7018 

7227 

7436 

7646 

7854 

209 

8 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938 

208 

9 

320146 

0354 

0562 

0769 

0977 

1  1184 

1391 

1598 

1805 

2012 

207 

210 

2219 

2426 

2633 

2839 

3046 

i  3252 

3458 

3665 

3871 

4077 

206 

1 

4282 

4488 

4094 

4899 

5105 

5310 

5516 

5721 

5926 

6131 

205    i 

2 

6336 

6541 

67'45 

6950 

7155 

7359 

7563 

7767 

7972 

8176 

204      : 

3 

8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

nnoft 

0211 

203 

4 

330414 

0617  i  0819 

1022  !  1225 

!  1427 

1630 

1832  !  2534 

2236 

202 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

225 

22.5 

45.0 

67.5 

90.0 

112.5 

135.0 

157.5 

180.0 

202.5 

224 

22.4 

44,8 

67.2 

89.6 

112.0 

134.4 

156.8 

179.2 

201.6 

223 

22.3 

44.6 

66.9 

89.2 

111.5 

133.8 

156.1 

178.4 

200.7 

222 

22.2 

44.4 

66.6 

88.8 

111.0 

133.2 

155.4 

177.6 

199.8 

221 

22.1 

44.2 

66.3 

88.4 

110.5 

132.6 

154.7 

176.8 

198.9 

220 

22.0 

44.0 

66.0 

88.0 

110.0 

132.0 

154.0 

176.0 

198.0 

219 

21.9 

43.8 

65.7 

87.6 

109.5 

131.4 

153.3 

175.2 

197.1 

218 

21.8 

43.6 

65.4 

87.2 

109.0 

130.8 

152.6 

174.4 

196.2 

217 

21.7 

43.4 

65.1 

86.8 

108.5 

130.2 

151.9 

173.6 

195.3 

216 

21.6 

43.2 

64.8 

86.4 

108.0 

129.6 

151.2 

172.8 

194.4 

215 

21.5 

43.0 

64.5 

86.0 

107.5 

129.0 

150.5 

172.0 

193.5 

214 

21.4 

42.8 

64.2 

85.6 

107.0 

128.4 

149.8 

171.2 

192.6 

213 

21.3 

42.6 

63.9 

85.2 

106.5 

127.8 

149.1 

170.4 

191.7 

212 

21.2 

42.4 

63.6 

84.8    i     106.0 

127.2 

148.4 

169.6 

190.8 

211 

21.1 

42.2 

63.3 

84.4 

105.5 

126.6 

147.7 

168.8 

189.9 

210 

21.0 

42.0 

63.0 

84.0 

105.0 

126.0 

147.0 

168.0 

189.0 

209 

20.9 

41.8 

62.7 

83.6 

104.5 

125.4 

146.3 

167.2 

188.1 

208 

20.8 

41.6 

62.4 

as.  2 

104.0 

124.8 

145.6 

166  4 

187.2 

207 

20.7 

41.4 

62.1 

82.8 

103.5       124.2 

144.9 

165.6 

186.3 

206 

20.6 

41.2 

61.8 

82.4 

103.0  i     123.6 

144.2       164.8 

185.4 

205 

20.5 

4tl  0 

C1.5 

82.0 

102.5 

123.0 

143.5 

164.0 

184.5 

204 

20.4 

40'.S 

61.2 

81.6 

102.0 

122.4 

142.8 

163.2 

183.6 

203    i  20.3 

40.6 

60.9 

81.2 

101.5 

121.8 

142.1  '     162.4 

182.7. 

202     i  20.2 

40.4 

60.6 

101.0 

121.2       141.4       161.6 

181.8 

TABLE    IX. — LOGARITHMS    OF   NUMBERS. 


No.  215  L.  332.]                                  [No.  239  L.  380. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

215 

332438 

2640 

2842 

3044 

3246 

3417 

3649 

3850 

4051 

4253 

202 

6 

4454 

4655 

4856 

5057 

5257 

5458 

5658 

5859 

6059 

6260 

201 

7 

G460 

6660 

6860 

7000 

7260 

7459 

7659 

7858 

8058 

8257 

200 

g 

8456 

8656 

8855 

9054 

9253 

9451 

9650 

9849 

0047 

0246 

1QQ 

9 

340444 

0642 

0841 

1039 

1237 

1435 

1632 

1830 

2028 

2225 

iyy 

198 

220 

2423 

2620 

2817 

3014 

3212 

3409 

3606 

3802 

3999 

4196 

197 

1 

4392 

4589 

4785 

4981 

5178 

5374 

5570 

5766 

5962 

6157 

196 

2 

6353 

6549 

6744 

6939 

7135 

7330 

7525 

7720 

7915 

8110 

195 

3 

8305 

8500 

8694 

8889 

9083 

9278 

9472 

9666 

9860 

0054 

1<V£ 

4 

350248 

0442 

0636 

0829 

1023 

1216 

1410 

1603 

1796 

1989 

1«74 

193 

5 

2183 

2375 

2568 

2761 

2954 

3147 

3339 

3532 

3724 

3916 

193 

6 

4108 

4301 

4493 

4685 

4876 

5068 

5260 

5452 

5643 

5834 

192 

7 

C026 

6217 

6408 

6599 

6790 

6981 

7172 

7363 

7554 

7744 

191 

8 

7935 

8125 

8316 

8506 

8696 

8886 

9076 

9266 

9456 

9646 

190 

9 

9835 

0025 

0215 

0404 

0593 

0783 

OQ75> 

1  1&1 

1  ^^in 

1  ^Q 

1ftO 

230 

361728 

1917 

2105 

2294 

2482 

2671 

\Ji)i4 

2859 

XiOl 

3048 

J.GOU 

3236 

JLOoV 

3424 

loy 

188 

1 

3612 

3800 

3988 

4176 

4363 

4551 

4739 

4926 

5113 

5301 

188 

2 

5488 

5675 

5862 

6049 

6236 

1  6423 

6610 

6796 

6983 

7169 

187 

3 

7356 

7542 

7729 

7915 

8101 

[  8287 

8473 

8659 

8845 

9030 

186 

4 

9216 

9401 

9587 

9772 

9958 

0143 

0328 

0513 

0698 

0883 

•JCK 

5 

371068 

1253 

1437 

1622 

1806 

1  1991 

2175 

2360 

2544 

2728 

1OO 

184 

6 

2912 

3096 

3280 

3464 

3647 

3831 

4015 

4198 

4382 

4565 

184 

7 

4748 

4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

183 

8 

6577 

67'59 

6942 

7124 

7306 

7488 

7670 

7852 

8034 

8216 

182 

9 

8398 

8580 

8761 

8943 

9124 

9306 

9487 

9668 

9849 

38 

0030 

181 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

202 
201 

20.2 
20.1 

40.4 
40.2 

60.6 
60.3 

80.8 
80.4 

101.0 

100.5 

121.2 
120.6 

141.4 
140.7 

161.6 
160.8 

181.8 
180.9 

200 

20.0 

40.0 

60.0 

80.0 

100.0 

120.0 

140.0 

160.0 

180.0 

199 

19.9 

39.8 

59.7 

79.6 

99.5 

119.4 

139.3 

159.2 

179.1 

198 

19  8 

39.6 

59.4 

79.2 

99.0 

118.8 

138.6 

158.4 

178.2 

197 

19.7 

39.4 

59.1 

78.8 

98.5 

118.2 

137.9 

157.6 

177.3 

196 

19  6 

39.2 

58.8 

78.4 

98  0 

117.6 

137.2 

156.8 

176.4 

195 

19.5 

39.0 

58.5 

78.0 

97.5 

117.0 

136.5 

156.0 

175.5 

194 

19.4 

38  8 

58.2 

77.6 

97.0 

116.4 

135.8 

155.2 

174.6 

193 

19  3 

38.6 

57.9 

77.2 

96.5 

115.8 

135.1 

154.4 

173.7 

192 

19  2 

38.4 

57.6 

76.8 

96.0 

115.2 

134.4 

153.6 

172.8 

li)l 

19.1 

38.2 

57.3 

76.4 

95.5 

114.6 

133.7 

152.8 

171.9 

190 

19.0 

38.0 

57.0 

76.0 

95.0 

114  0 

133.0 

152.0 

171.0 

189 

18.9 

37.8 

56.7 

75.6 

94  5 

113.4 

132.3 

151.2 

170.1 

188 

18  8 

37.6 

56.4 

75.2 

94.0 

112.8 

131.6 

150.4 

169.2 

187 

18.7 

374 

56.1 

74.8 

93.5 

112.2 

130.9 

149.6 

168.3 

186 

18.6 

37.2 

55.8 

74.4 

93.0 

111.6 

130.2 

148.8 

167.4 

185 

18  5 

37.0 

55  5 

74.0 

92  5 

111.0 

129.5 

148.0 

166.5 

184 

18  4 

36  8 

55.2 

73.6 

92.0 

110.4 

128.8 

147.2 

165.6 

183 

183 

36  6 

54.9 

73.2 

91.5 

109  8 

128.1 

146  4 

164.7 

182 

182 

36.4 

54  6 

72  8 

91  0 

109  2 

127.4 

145.6 

163.8 

181 

18  1 

36  2 

54  3 

72  4 

90.5 

108  6 

126.7 

144.8 

162.9 

180 

18  0 

36  0 

54  0 

72  0 

90  0 

108  0 

126  0 

144.0 

162.0 

179 

17  9 

35  8 

53.7 

71.6 

89  5 

107.4 

125.3 

143.2 

161.1 

TABLE    IX. — LOGARITHMS    OF    NUMBERS. 


No.  240  L.  380.] 

jNo,  269  L.  431. 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Diff. 

240 

380211 

0392 

0573 

0754 

0934 

1115 

1296     1476 

1656 

1837 

181 

1 

2017 

2197 

2377 

2557 

2737 

2917 

3097     3277 

3456 

3636 

180 

2 

3815 

3995 

4174 

435 

3 

45:33 

4712 

4891 

5( 

)70 

5249 

5428 

179 

3 

5606 

5785 

5964 

614 

2 

6321 

6499 

6677     6, 

S.-56 

70134 

7212 

178 

4 

7390 

7568 

7746 

7924 

8101 

8279 

8456     86:34 

8811 

8989 

178 

OKlft 

9520 

969 

a      oar:; 

5 

ulOO 

jo-to 

0051 

022S      n 

in" 

0582 

0759 

177 

6 

390935 

1112 

1288 

1464      1641 

1817 

1993 

2169 

2345 

2521 

176 

7 

2697 

2873 

3048 

322  1      3400 

3575 

3751 

3926 

4101 

4277 

176 

8 

4452 

4627 

4802 

497 

7      5152 

5326 

5501 

5< 

J76 

5850 

6025 

175 

9 

6199 

6374 

6548 

6722 

6896 

7071 

7245 

7419 

7592 

7766 

174, 

•250 
1 

7940 
9674 

8114 
9847 

8287     8461     8634 

8808 

8981 

9154 

9328 

9501 

173 

i 

0020 

0192 

0365 

0538 

0711 

o 

SKM 

1056 

1228 

173 

2 

401401 

1573 

1745 

1917 

2089 

2261 

2433 

2605 

2777 

2949 

172 

3 

3121 

3292 

3464 

36J. 

5 

.3807 

!  3978 

4149 

4 

320 

4492 

4663 

171 

4 

4834 

5005 

5176 

534 

6 

5517 

5688 

5858 

6 

)29 

6199 

6370 

171 

5 

6540 

6710 

6881 

7051 

7221 

7391 

7561 

731 

7901 

8070 

170 

6 

8240 

8410 

8579 

874 

9 

8918 

9087 

9257 

0 

42(5 

9595 

9764 

109 

7 

9933 

I 

0102 

0271 

0440 

0609 

0777 

0046 

1114 

1283 

1451 

169 

8 

411620 

1788 

1956 

2124 

2293 

i  2461 

2(521) 

2796 

2964 

3132 

168 

9 

3300 

3467 

3635 

3803 

3970 

4137 

4305 

4472 

4039 

4806 

167 

260 

4973 

5140 

5307 

54; 

4 

5641 

5808 

5974 

6141 

6308 

6474 

167 

1 

6(541 

6807 

6973 

71: 

9 

7306 

7472 

7638 

7 

sol 

7970 

8135 

166 

2 
3 

8301 
9956 

8467 

86:33     8798 

8964 

9129 

9295 

9460     9625 

9791 

165 

0121 

0286 

0451 

0616 

0781     0945 

1110     1275 

1439 

165 

4 

421604 

1768 

19:33 

20< 

)7 

2261 

2426 

2590 

2 

75  1 

2918 

3082 

164 

5 

3246 

3410 

3574 

3737 

3901 

4065 

4228 

4392 

4555 

4718 

164 

6 

4882 

5045 

5208 

53' 

*1 

5534 

501)7 

5860 

6 

023 

6186 

6:349 

163 

7 

6511 

6674 

6836 

691 

)1) 

7161 

7324 

7486 

7 

518 

7811 

7973 

162 

8 

8135 

8297 

8459 

8621 

8783 

8944 

9106 

9268 

9429 

9591 

162 

g 

9752 

9914 

43 

0075     0236     0398 

0559 

0720     0881 

1042 

1203 

161 

PROPORTIONAL  PARTS. 

Diff 

178 

1 

2 

35.6 

3 

4 

5 

6 

106.8 

7             8 

9 

17.8 

53.4 

71.2 

89.0 

124.6       142.4 

160.2 

177 

17.7 

35.4         53.1 

70.8 

88.5 

106.2 

123.9       141.6 

159.3 

176 

r.a 

35.2         52.8 

70.4 

88.0 

105.6 

123.2       140.8 

158.4 

175 

r.s 

35.0         52.5 

70.0 

87.5 

105.0 

122.5       140.0 

157.5 

174 

11.4 

34.8         52.2 

69.6 

87.0 

104.4 

121.8       139.2 

156.6 

173 

17.8 

34.6         51.9 

69.2 

86.5 

103.8 

121.1       138.4 

155.7 

172 

17.2 

34.4         51.6 

68.8 

86.0 

103.2 

120.4       137.6 

154.8 

171 

17.1 

34.2         51.3 

68.4 

85.5 

102.6 

119.7       136.8 

153.9 

170 

r.o 

34.0         51.0 

68.0 

85.0 

102.0 

119.0       136.0 

153.0 

169 

16.9 

33.8         50.7 

67.6 

84.5 

101.4 

118.3       135.2 

152.1 

168 

16.8 

33.6         50.4 

67  2 

84.0 

100.8 

117.6       134.4 

151.2 

167 

16.7 

33.4         50.1 

66  .'8 

83.5 

100.2 

116.9       133.6 

150.3 

106 

16.6 

33.2         49.8 

66.4 

83.0 

99.6 

116.2       132.8 

149.4 

165 

16.5 

33.0         49.5 

66.0 

82.5 

99.0 

115.5       132.0 

148.5 

164 

16.4 

32.8         49.2 

65.6 

82.0 

98.4 

114.8       131.2 

147.6 

163 

16.3 

32.6         48.9 

65.2 

81.5 

97.8 

114.1       130.4 

146.7 

16.2 

32.4         48.5 

64.8 

81.0 

97.2 

113.4       129.6 

145.8 

161 

16.1 

32.2         48.3 

64.4 

80.5 

96.6 

112.7       128.8 

144.9 

85 


TABLH    IX. — LOGARITHMS    OF    XUMBERS. 


No.  270  L.  431.] 

[No.  299  L.  476. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

270 

431364 

1525 

1685 

1846 

2007 

2167 

2328 

2488 

2649 

2809 

161 

1 

2969 

3130 

3290 

345 

0 

3610 

3770 

3930 

4090 

4249 

4409 

160 

2 

4569 

4729 

4888 

504 

8 

5207 

5367 

5526 

5685 

5844 

6004 

159 

3 

6163 

6322 

6481 

6640 

6799 

6957 

7116 

7275 

7433 

7592 

159 

4 

7751 

7909 

8067 

822 

I) 

8384 

8542  8701 

8859 

9017 

9175 

158 

5 

9333 

9491 

9648 

9806 

9964 

0122  ra>7o 

0437 

0594 

0752 

•JKQ 

6 

440909 

1066 

1224 

1381 

1538 

1695 

1852 

2009 

2166 

2323 

1OO 

157 

7 

2480 

2637 

2793 

2950 

3106 

3263 

3419 

3576 

3732 

3889 

157 

8 

4045 

4201 

4357 

451 

3 

4669 

4825 

4981 

5137 

5293 

5449 

156 

9 

5604 

5760 

5915 

6071 

6226 

6382 

6537 

6692 

6848 

7003 

155 

280 

7158 

7313 

7468 

7623 

7778 

79.33 

8088 

8242 

8397 

8552 

155 

1 

8706 

8861 

9015 

9170 

9324 

9478 

9633 

9787 

9941 

0095 

154 

2 

450249 

0403 

0557 

0711 

0865 

1018 

1172  i  1326 

1479 

1633 

154 

3 

1786 

1940 

2093 

224 

[7 

2400 

2553 

2706 

2859 

3012 

3165 

153 

4 

3318 

3471 

3624 

3777 

3930 

4082 

4235 

4387 

4540 

4692 

153 

5 

4845 

4997 

5150 

53C 

2 

5454 

5606 

5758 

5910 

6062 

6214 

152 

6 

6366 

6518 

6670 

68* 

1 

6973 

7125 

727'6 

7428 

7579 

7731 

152 

7 

7882 

8033 

8184 

8336 

8487 

8638 

8789 

8940 

9091 

9242 

151 

g 

9392 

9543 

9694 

984 

5 

9995 

0146 

0296  '•  n/M'7 

0597 

O^dft 

9 

460898 

1048 

1198 

1348 

1499 

1649 

1799 

1948 

2098 

2248 

150 

290 

2398 

2548 

2697 

2847 

2997 

3146 

3296 

3445 

3594 

3744 

150 

1 

3893 

4042 

4191 

434 

0 

4490 

4639 

4788 

4936 

5085 

5234 

149 

2 

5383 

5532 

5680 

5829 

5977 

6126 

6274 

6423 

6571 

6719 

149 

3 

6868 

7016 

7164 

731 

2 

7460 

7608 

7756 

7904 

8052 

8200 

148 

4 

8347 

8495 

8643 

8790 

8938 

9085  9233 

9380 

9527 

9675 

148 

5 

9822 

9969 

0116 

02G 

Q 

0410 

0557 

0704 

0851 

0998 

1145 

147 

6 

471292 

1438 

1585 

1732 

1878 

2025 

2171 

2318 

2464 

2610 

111 

146 

7 

2756 

2903 

3049 

31£ 

5 

3341 

3487 

3033 

3779 

3925 

4071 

146 

8 

4216 

4362 

4508 

4653 

4799 

4944 

5090 

5235 

5381 

5526 

146 

9 

5671 

5816 

5962 

6107 

6252 

6397  6542 

6087 

6832 

6976 

145 

I 

PROPORTIONAL  PARTS. 

Diff.   1 

2 

3 

4 

5 

6      7 

8 

9 

161   16.1 

32.2 

48.3 

64.4 

80.5 

96.6    112.7 

128.8 

144.9 

160   16.0 

32.0 

48.0 

64.0 

80.0 

96.0    112.0 

128.0 

144.0 

15»   15.9 

31.8 

47.7 

63.6 

79.5 

95.4    111.3 

127.2 

143.1 

158   15.8 

31.6 

47.4 

63.2 

79.0 

94.8    110.6 

126.4 

142.2 

157  1  15.7 

31.4 

47.1 

62.8 

78.5 

94.2    109.9 

125.6 

141.3 

156   15.6 

31.2 

46.8 

62.4 

78.0 

93.6    109.2 

124.8 

140.4 

155   15.5 

31.0 

46.5 

62.0 

77  .  5 

93.0    108.5 

124.0 

139.5 

154   15.4 

30.8 

46.2 

61.6 

77.0 

92.4    107.8 

123.2 

138.6 

153   15.3 

30.6 

45.9 

61.2 

76.5 

91.8    107.1 

122.4 

137.7 

152   15.2 

30.4 

45.6 

60.8 

76.0 

91.2    106.4 

121.6 

136.8 

151   15.1 

30.2 

45.3 

60.4 

75.5 

90.6    105.7 

120.8 

135.9 

150   15.0 

30.0 

45.0 

60.0 

75.0 

90.0    105.0 

120.0 

135.0 

149   14.9 

29.8 

44.7 

59.6 

74.5 

89.4    104.3 

119.2 

134.1 

148   14.8 

29.6 

44.4 

59.2 

74.0 

88.8    103.6 

118.4 

133.2 

147   14.7 

29.4 

44.1 

58.8 

73.5 

88.2    102.9 

117.6 

132.3 

146   14.6 

29.2 

43.8 

58.4 

73.0 

87.6    102.2 

116.8 

131.4 

145   14.5 

29.0 

43.5 

58.0 

72.5 

87.0    101:5 

116.0 

130.5 

144   14.4 

28.8 

43.2 

57.6 

72.0 

86.4    100.8 

115.2 

129.6 

143   14.3 

28.6 

42.9 

57.2 

71.5 

85.8  1  100.1 

114.4 

128.7 

142   14.2 

28.4 

42.6 

56.8 

71.0 

85  2    99.4 

113.6 

127.8 

141   14.1 

28.2 

42.3 

56.4 

70.5 

84.6    98.7 

112.8 

126.9 

140   14.0 

28.0 

42.0 

56.0 

70.0 

84.0     98.0 

112.0 

126.0 

TABLE    IX. — LOGARITHMS    OF    NUMBERS. 


No.  300  L.  477.] 

[No.  339  L.  531. 

N. 

0 

1 

2 

8 

4 

5 

6 

7 

8 

9 

Diff. 

300 
1 

2 

3 
4 
5 
6 

7 
8 
9 

310 
1 
2 
3 
4 
5 
6 

8 
9 

320 
1 
o 

3 

4 
5 
6 
7 
8 
9 

330 
1 

2 
3 
4 
5 
6 
7 
8 

9 

477121 
8566 

7266 
8711 

7411 
8855 

7555 
8999 

7700 
9143 

7844 
9287 

7989 
9431 

8133 
9575 

8278 
9719 

1156 
2588 
4015 
5437 
6855 
8269 
9677 

R4^ 
9863 

145 
144 

144 
143 
143 
142 
142 
141 
141 

140 

140 
139 
139 
139 
138 
138 

137 
137 
136 
136 

136 
135 
135 

134 
134 
133 
133 
133 
132 
132 

131 

131 
131 
130 
130 
129 
129 
129 

128 
128 

480007 
1443 
2874 
4300 
5721 
7138 
8551 
9958 

0151 
1586 
3016 
4442 
5863 
7280 
8692 

0294 
1729 
3159 
4585 
6005 
7421 
8833 

0438 
1872 

33<X> 
4727 
6147 
7563 
8974 

0582 
2016 
13445 
4869 
6289 
7704 
9114 

0725 
2159 
3587 
5011 
6430 
7845 
9255 

0869 
2:302 
3730 
5153 
6572 
7986 
9396 

1012 
2445 
3872 
5295 
6714 
8127 
9537 

1299 
2731 
4157 
5579 
6997 
8410 
9818 

0099 

1502 
2900 
4294 

seas 

7068 
8448 
9824 

0239 

1642 
3040 
44:33 

5822 
7206 
8586 
9962 

0380 

1782 
3179 
4572 
5960 
7344 
8724 

0520 

1922 
3319 
4711 
6099 

7483 
8862 

i  0661 

2062 
3458 
4850 
62:38 
7621 
1  8999 

0801 

2201 
3597 
4989 
6376 
7759 
9137 

0941 

2341 
3737 
5128 
6515 
7897 
9275 

1061 
2481 
3876 
5267 
6653 
8035 
9412 

1222 

2621 
4015 
5406 
6791 
8173 
9550 

491362 
2760 
4155 
5544 
6930 
8311 
9687 

0099 
1470 
2837 
4199 

5557 
6911 

&J60 
9006 

0236 
1607 
2973 
4335 

5693 
7046 

8395 
9T40 

1081 
2418 
3750 
5079 
6403 
7724 

9040 

0374 
1744 
3109 
4471 

5828 
!  7181 
85:30 
9874 

0511 
1880 
3246 
4607 

5964 
7316 
8664 

0648 
2017 
3382 
4743 

6099 
7451 
8799 

0785 
2154 
3518 
4878 

6234 
7586 
8934 

0922 
2291 
3655 
5014 

6370 
7721 

(!068 

501059 
2427 
3791 

5150 
6505 

7856 
9203 

510545 

1883 
3218 
4548 
5874 
7196 

8514 
9828 

521138 
2444 
3746 
5045 
6339 
7630 
8917 

"530200" 

1196 
2564 
3927 

5286 
6640 
7991 
9337 

1333 

2700 
4063 

5421 
6776 

8126 
9471 

0009 
1349 
2684 
4016 
5344 
6668 
7987 

9303 

0615 
1922 
3226 
4526 
5822 
7114 
8402 
9687 

0143 
1482 
2818 
4149 
5476 
6800 
8119 

9434 

0745 
2053 
3356 
4656 
5951 
7243 
8531 
9815 

IC96 

0277 
1616 
2951 
4282 
£609 
6932 
8251 

9566 

0876 
2183 
3486 
4785 
C081 
7372 

ecco 

9943 
1223 

0411 
1750 
S084 
4415 
5741 
7064 
8282 

9697 

1C07 
2314 
3616 
4915 
C210 
7^01 
8788 

0679 
2017 
3351 

4681 
6006 
7328 

8646 
9959 

0813 
2151 
3484 
4813 
6139 
7460 

8777 

0947 
2284 
3617 
4946 
6271 
7592 

8909 

1215 
2551 

3883 
5211 
6535 
7855 

9171 

0090 
1400 
2705 
4006 
5304 
6598 
7888 
9174 

0221 

1530 
2835 
4136 
54:34 
6727 
8016 
9302 

0353 
1661 
2966 
4266 
5563 
6856 
8145 
9430 

i  0-184 
1792 
3096 
4396 
i  5693 
6985 
1  8274 
9559 

1269 
2575 
3876 
5174 
6469 
7759 
9045 

0072 
1351 

0328 

0456 

0584 

0712 

0840  0968 

PROPORTIONAL  PARTS. 

Diff.   1 

2     3 

4 

5 

6 

7 

8 

9 

139   13.9 
138   13.8 
137   13.7 
136   13.6 
1:55   13.5 
134   13.4 
133   13.3 
132   13.2 
131   13.1 
130   13.0 
129   12.9 
128   12.8 
127   12  7 

2".  8    41.7 
2".  6    41.4 
2~.4    41.1 
2".  2    40.8 
2~.0    40.5 
26.8    40.2 
26.6    39.9 
26.4    39.6 
26.2    89.3 
26.0    89.0 
25.8    38.7 
25.6    38.4 
25.4    38.1 

55.6 
55.2 
&4.8 
54.4 
54.0 
53.6 
53.2 
52.8 
52.4 
52.0 
51.6 
51.2 
50.8 

69.5 
69.0 
68.5 
68.0 
67.5 
67.0 
66.5 
66.0 
65.5 
65.0 
64.5 
64.0 
63.5 

83.4 

82.8 
82.2 
81.6 
81.0 
80.4 
79.8 
79.2 
78.6 
78.0 
77.4 
76.8 
76.2 

97.3 
96.6 
95.9 
95.2 
94.5 
93.8 
93.1 
92.4 
91.7 
91.0 
90.3 
89.6 
88.9 

111.2 
110.4 
109.6 
108.8 
108.0 
107.2 
106.4 
105.6 
104.8 
104.0 
103.2 
102.4 
101.6 

125.1 
124.2 
123.3 
122.4 
121.5 
120.6 
119.7 
118.8 
117.9 
117.0 
116.1 
115.2 
114.3 

TABLE    IX. — LOGARITHMS    OF    NUMBERS. 


No.  340  L.  531.] 

[No.  379  L.  579. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

340 
1 
2 

3 
4 
5 
6 

8 
9 

350 
1 
2 

3 

4 

5 
6 

8 
9 

360 
1 
2 
3 

4 
5 

6 

7 
8 
9 

370 
1 

2 

3 
4 
5 

6 
7 
8 
9 

531479 
2754 
4026 
5294 
6558 
7819 
9076 

1607 
2882 
4153 
5421 
6685 
7945 
9202 

1734 

3009 
4280 
5547 
6811 
8071 
9327 

1862 
3136 
4407 
5674 
6937 
8197 
9452 

1990 
3264 
4534 
5800 
7063 
8322 
9578 

2117 
3391 
4661 
!  5927 
7189 
8448 
9703 

2245 
3518 
4787 
6053 
7315 
8574 
9829 

2372 
3645 
4914 
6180 
7441 
8699 
9954 

1205 
2452 
3096 

4936 
6172 
7405 
8635 
9861 

2500 
3772 
5041 
6306 
7567 
8825 

2627 
3899 
5167 
6432 
7693 
8951 

128 
127 
127 
126 
126 
126 

125 
125 
125 
124 

124 
124 
123 
123 

123 
122 
122 
121 
121 
121 

120 
120 
120 

119 
119 
119 
119 
118 
118 
118 

117 

/1  7 
117 
116 
116 
116 
115 
115 
115 
114 

0079 
1330 
2576 

3820 

5060 
6296 
7529 
8758 
9984 

0204 
1454 
2701 
3944 

5183 
6419 
7652 

8881 

540329 
1579 
2825 

4068 
5307 
6543 

7775 
9003 

0455 
1704 
2950 

4192 
5431 
6666 

7898 
9126 

0580 
1829 
3074 

4316 
5555 

6789 
8021 
9249 

0705 
1953 
3199 

4440 

5678 
6913 
8144 
9371 

0830 
207'8 
3323 

4564 
5802 
7036 
8267 
9494 

0955 
i  2203 
1  3447 

4688 
5925 
7159 

8389 
9616 

1080 
2327 
3571 

4812 
6049 
7282 
8512 
97'39 

0106 
1328 
2547 
3762 
4973 
6182 

7387 
8589 
9787 

550228 
1450 
2668 
3883 
5094 

6303 
7507 
8709 
9907 

0351 
1572 
2790 
4004 
5215 

6423 

7627 

8829 

0473 
1694 
2911 
4126 
5336 

6544 
7748 
8948 

0595 
1816 
3033 
4247 
5457 

6664 

7868 
9068 

0717 
1938 
3155 
4368 
5578 

6785 

7988 
9188 

;  0840 
2060 
3276 
4489 
5699 

6905 
8108 
9308 

0962 
2181 
3398 
4610 
5820 

7026 
8228 
9428 

1084 
2303 
3519 
4731 
5940 

7146 
8349 
9548 

1206 
2425 
3640 
4852 
6061 

7267 
8409 
9667 

0026 
1221 
2412 
3600 
4784 
5966 
7144 

8319 
9491 

0146 
1340 
2531 
3718 
4903 
6084 
7262 

8436 
9608 

0265 
1459 
2650 
3837 
5021 
6202 
7379 

8554 
9725 

0385 
1578 
2769 
3955 
5139 
6320 
7497 

8671 
9842 

i  0504 
!  1698 
2887 
4074 
5257 
6437 
7614 

8788 
9959 

0624 
1817 
3006 
4192 
5376 
6555 
7732 

8905 

0743 
1936 
3125 
4311 
5494 
6673 
7849 

9023 

0863 
2055 
3244 
4429 
5612 
6791 
7967 

9140 

0982 
2174 
3362 
4548 
5730 
6909 
8084 

9257 

561101 
2293 
3481 
4666 
5848 
7026 

8202 
9374 

0076 
1243 

2407 
3568 
47'26 
5880 
7032 
8181 
9326 

0193 
1359 
2523 
3684 
4841 
5996 
7147 
8295 
9441 

0309 
1476 
2639 
3800 
4957 
6111 
7262 
8410 
9555 

0426 
1592 
2755 
3915 
5072 
6226 
7377 
8525 
9669 

570543 
1709 
2872 
4031 
5188 
6341 
7492 
8639 

0660 
1825 
2988 
4147 
5303 
6457 
7607 
8754 

0776 
1942 
3104 
4263 
5419 
6572 
7722 
8868 

0893 
2058 
3220 
4379 
5534 
6687 
7836 
8983 

1010 
2174 
3336 
4494 
5650 
6802 
7951 
9097 

i  1126 
2291 
3452 
4610 
5765 
6917 
8066 
9212 

PROPORTIONAL  PARTS. 

Dift   1 

2      3 

4 

5 

64.0 
63.5 
63.0 
62.5 
62.0 
61.5 
61.0 
60.5 
60.0 
5(J.5 

6      7 

8 

9 

115.2 
114.3 
113.4 
112.5 
111.6 
110.7 
109.8 
108.9 
108.0 
107.1 

128   12.8 
127   12  7 
126   12  6 
125   12.5 
124   12.4 
123   12.3 
122   12.2 
121   12.1 
120   12.0 
119   11  9 

25.6    38.4 
25  4    38.1 
25.2    37.8 
25.0    37.5 
24.8    37.2 
24.6    36.9 
244    36.6 
24.2    36.3 
24  0    360 
23.8    35.7 

51.2 
50.8 
50.4 
50.0 
49.6 
49.2 
48.8 
48.4 
48.0 
47.6 

76.8    89.6 
76.2    88.9 
75.6    88.2 
75.0    87.5 
74.4    86  8 
73.8    86.1 
73.2    85.4 
72.6    84.7 
72.0    84.0 
71.4    83.3 

102.4 
101.6 
100.8 
100.0 
99.2 
98.4 
97.6 
96.8 
96.0 
95.2 

TABLE    IX. — LOGARITHMS    OF    NUMBERS. 


No.  380.  L.  579.] 

[No.  414  L.  617. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

380 

579784 

9898 

0012 

0126 

0241 

0355 

0469 

0583 

0697 

0811 

114 

1 

580925 

1039 

1153 

1207 

1381 

1495 

1608 

17 

22 

1830 

1950 

2063 

2177 

2291 

2404 

2518 

9681 

2745 

2858 

2972 

3085 

3 

3199 

&312 

3426 

3539 

3652 

3705 

3879 

31 

92 

4105 

4218 

4 

4331 

4444 

4557 

4070 

4783 

4896 

5009 

5122 

5235 

5348 

113 

5 

5401 

5574 

5080 

5799 

5912 

6004 

0137 

6; 

'50 

6362 

6475 

0 

6587 

6700 

OS  12 

6925 

7037 

7149 

7262 

71 

174 

7488 

7599 

7 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

8008 

8720 

112 

8 

8832 
9950 

8944 

9050 

9167 

9279 

9391 

9503     9015 

9720 

9838 

0061 

0173 

0284 

0396 

0507 

0619 

0730 

0842 

0953 

300 

591005 

1176 

1287 

1399 

1510 

1021 

1732 

1843 

1955 

2066 

1 

2177 

2288 

2399 

8510 

2621 

2732 

0843 

2' 

)54 

3004 

3175 

111 

2 

3286 

3397 

8606 

301S 

3729 

3840 

3950 

4001 

4171 

4282 

3 

4393 

4503 

4014 

4724 

4834 

4945 

5055 

5 

65 

5876 

5380 

4 

5496 

5000 

5717 

5827 

5937 

6047 

6157 

65 

J67 

0377 

01S7 

5 

0597 

6707 

OS  17 

0927 

7037 

7146 

7256 

7306 

7476 

7588 

110 

6 

7095 

7805 

7914 

8024 

8134 

8243 

8868 

8402 

8572 

8081 

7 

8791 

()Ui>-> 

8900 

9009 

9119 

9228 

9337 

9440 

9556 

9665 

9774 

0101 

0210 

0319 

0428  !  0537 

0046 

0755 

0861 

109 

9 

000973 

1082 

1191 

1299 

1408 

1517 

1025 

1 

84 

1813 

1951 

400 

2000 

2109 

2277 

2386 

2494 

2603 

2711 

2819 

2928 

3036 

1 

8144 

3253 

3:501 

34(59 

&577 

8686 

8794 

3902 

4010 

4118 

108 

0 

4220 

4334 

4142 

4550 

4658 

4706 

4874 

4< 

)S2 

r>oh9 

5197 

8 

13306 

5413 

5521 

5626 

573(5 

5844 

5951 

6 

159 

6100 

0274 

4 

6381 

6489 

5596 

0701 

6811 

6919 

7020 

7133 

72  \  1 

7348 

5 

7455 

7562 

7660 

7777 

7884 

7991 

8098 

8 

.'05 

8312 

8419 

107 

6 

8526 

<l"Vl  1 

8033 

Q^fll 

8740 

')SOS 

8847 

Qfll   4 

8954 

9001 

9167 

9274 

9381 

9  188 

' 

00°  1 

01  °8 

0234 

0311 

0447 

0554 

8 

010600 

0767 

0873 

0979 

1086 

1192 

1298 

1  11)5 

loll 

1017 

9 

1723 

1829 

1936 

2042 

2148 

2254 

2300 

2400 

2572 

2078 

106 

410 

2784 

2890 

2996 

3102 

3207 

3313 

3419 

3525 

3030 

3736 

1 

3842 

8947 

4053 

4159 

4264 

4370 

4475 

4 

•)81 

4080 

4792 

2 

4897 

5003 

5108 

5213 

Ml  9 

5424 

5529 

5 

134 

5740 

5845 

8 

5950 

6055 

6160 

62(55 

6370 

6476 

6581 

6086 

0790 

6895 

105 

4 

7000 

7105 

7210 

7315 

7420 

7525 

7629     7 

734 

783.) 

7943 

PROPORTIONAL  PARTS. 

Diff.       1 

234 

5 

6 

7 

8 

9 

118       11.8 

23.6         35.4         47.2 

59.0 

70.8 

82.0 

94.4 

106.2 

117       11.7 

23.4         35.1         40.8 

58.5 

70.2 

81.9 

93.6 

1D5.3 

116       11.6 

23.2         34.8         46.4 

58.0 

69.6 

81.2 

92.8 

104.4 

115       11.5 

23.0         34.5         40.0 

57.5 

69.0 

80.5 

92.0 

103.5 

114       11.4 

22.8         34.2         45.0 

57.0 

68.4 

79.8 

91.2 

102.0 

113       11.3 

22.6         33.9         45.2 

56.5 

67.8 

79.1 

90.1 

101.7 

112       11.2 

22.4        33.6        44.8 

56.0 

67.2 

78.4 

89.6 

100.8 

111       11.1 

22.2         33.3         44.4 

55.5 

60.0 

77.7 

88.8 

99.9 

110       11.0 

22.0         33.0         44.0 

55.0 

66.0 

77.0 

88.0 

99.0 

109       10.9 

21.8         32.7         43.6 

54.5 

65.4 

70.3 

87.2 

98.1 

108       10.8 

21.6         32.4         43.2 

54.0 

64.8 

75.0 

86.4 

97.x! 

107       10.7 

21.4         32.1         42.8 

53.5 

64.2 

74.9 

&5.6 

96.3 

106       10.6 

21.2         31.8         42.4 

53.0 

63.6 

74.2 

84.8 

95.4 

105       10.5 

21.0        31.5        42.0 

52.5 

63.0 

73.5 

84.0 

94.5 

105       10.5 

21.0         31.5         42.0 

52.5 

63.0 

73.5 

84.0 

94.5 

104       10.4 

20.8         31.2         41.6 

52.0 

62.4 

72.8 

83.2 

93.6 

89 


TABLE    IX. — LOGARITHMS    OF    NUMBERS. 


No.  415  L.  618.]                                   [No.  459  L.  662 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Diff. 

415 
6 

8 
9 

420 
1 
2 
3 
4 
5 
6 

8 
9 

430 
1 
2 
3 

4 
5 
6 

7 
8 
9 

440 
1 
2 

3 

4 

6 

7 
8 
9 

450 
1 
2 
3 

4 
5 
6 

7 

8 
9 

618048 
9093 

8153 
9198 

8257 
9302 

8362 
9406 

8466 
9511 

8571 
9615 

8676 
9719 

0760 
1799 
2835 

3869 
4901 
5929 
6956 
7980 
9002 

8780 
9824 

8884 
9928 

8989 

0032 
1072 
2110 
3146 

4179 
5210 

6238 
7263 
8287 
9308 

105 
104 

103 
102 

101 
100 

99 
98 

97 
96 

95 

620136 
1176 
2214 

3249 

4282 
5312 
6340 
7366 
8389 
9410 

630428 
1444 
2457 

3468 
4477 
5484 
6488 
7490 
8489 
9486 

640481 
1474 
2465 

3453 

4439 
5422 
6404 
7383 
8360 
9335 

0240 
1280 
2318 

3353 

4385 
5415 
6443 
7468 
8491 
9512 

0530 
1545 
2559 

3569 
4578 
5584 
6588 
7590 
8589 
9586 

0344 

1384 
2421 

3456 
4488 
5518 
6546 
7571 
8593 
9613 

0631 
1647 
2660 

3670 
4679 

5685 
6688 
7690 
8689 
9686 

0448 
1488 
2525 

3559 
4591 
5621 
6648 
7673 
8695 
9715 

0783 

1748 
2761 

3771 
4779 

5785 
6789 
7790 
8789 
.9785 

0552 
1592 
2628 

3663 
4695 
5724 
6751 

7775 
8797 
9817 

0835 
1849 
2862 

3872 
4880 
5886 
6889 
7890 
8888 
9885 

!  0656 
1695 
2732 

3766 
4798 

5827 
6853 
787'8 
8900 
9919 

0936 
1951 
2963 

3973 

4981 
5986 
6989 
7990 
8988 
9984 

0864 
1903 
2939 

3973 

5004 
6032 
7058 
8082 
9104 

0968 
2007 
3042 

4076 
5107 
6135 
7161 

8185 
9206 

0021 
1038 
2052 
3064 

4074 
5081 

6087 
7089 
8090 
9088 

0123 
1139 
2153 
3165 

4175 

5182 
6187 
7189 
8190 
9188 

0224 
1241 
2255 
3266 

4276 
5283 
6287 
7290 
8290 
9287 

0326 
1342 
2356 
3367 

4376 

5383 
6388 
7390 
8389 
9387 

0084 
1077 
2069 
3058 

4044 
5029 
6011 
6992 
7969 
8945 
9919 

0183 
1177 
2168 
3156 

4143 
5127 
6110 

7089 
8067 
9043 

~ooieT 

0987 
1956 
2923 

3888 
4850 
5810 
6769 
7725 
8679 
9631 

0581 
1529 
2475 

0283 
1276 
2267 
3255 

4242 
5226 
6208 
7187 
8165 
9140 

0113 
1084 
2053 
3019 

3984 
4946 
5906 
6864 
7820 
8774 
9726 

0676 
1623 
2569 

0382 
1375 
2366 
3354 

4340 
5324 
6306 

7285 
8262 
9237 

0210 
1181 
2150 
3116 

4080 
5042 
6002 
6960 
7916 
8870 
9821 

0581 
1573 
2563 

3551 

4537 
5521 
6502 
7481 
8458 
9432 

0680 
1672 
2662 

3650 
4636 
5619 
6600 
7579 
8555 
9530 

0779 
1771 
2761 

3749 
4734 
5717 

6698 
7676 
8653 
9627 

0879 
1871 
2860 

3847 
4832 
5815 
6796 
7774 
8750 
9724 

0978 
1970 
2959 

3946 
4931 
5913 
6894 

7872 
8848 
9821 

650308 
1278 
2246 

3213 
4177 
5138 
6098 
7056 
8011 
8965 
9916 

0405 
1375 
2343 

3309 
4273 
5235 
6194 
7152 
8107 
9060 

0011 
0960 
1907 

0502 
1472 
2440 

3405 
4369 
5331 
6290 

7247 
8202 
9155 

0599 
1569 
2536 

3502 
4465 
5427 
6386 
7343 
8298 
9250 

0696 
1666 
2633 

3598 
4562 
5523 
6482 
7438 
8393 
9346 

0793 
1762 
2730 

3695 
4658 
5619 
6577 
7534 
8488 
9441 

0890 
1859 
2826 

3791 

4754 
5715 
6673 
7629 
8584 
9536 

0486 
1434 
2380 

0106 
1055 
2002 

0201 
1150 
2096 

0296 
1245 
2191 

0391 
1339 

2286 

0771 
1718 
2663 

660865 
1813 

PROPORTIONAL  PARTS. 

Diff,   1 

2 

21.0 
20  8 
20  6 
20  4 
20  2 
20  0 
19  8 

3      4 

5 

678 

63  0    73.5    84  0 
624    728    832 
618    721    82.4 
61  2    71  4    81  6 
60  6    70  7    80  8 
60.0    70  0    80  0 
59  4    69  3    79  2 

9 

94.5 
93.6 
92  7 
91  8 
90.9 
90  0 
89  1 

105   10  5 
104   10  4 
103   103 
102   10  2 
101   10  1 
100   10.0 
99    99 

31  5    42.0 
bl  2    41  6 
309    41.2 
30  6    40  8 
30  3    40.4 
30.0    40  0 
29  7    39  6 

52  5 
52.0 
51  5 
51  0 
50  5 
50  0 
49  5 

90 


TABLE    IX. — LOGARITHMS    OF    NUMBERS. 


No.  460  L.  662.] 

[No.  499  L.  698. 

N. 

0 

1 

2 

8 

4 

5 

6 

7 

8 

9 

Diff. 

460 

662758 

2852 

2947 

3041 

3135 

3230 

3324 

3418 

3512 

3607 

1 

3701 

3795 

3889 

39fc 

<3 

4078 

4172 

4266 

4 

300 

445- 

i 

4548 

2 

4642 

4736 

4830 

4924 

5018 

5112 

5206 

5299 

5393 

5487 

94 

3 

5581 

5675 

5769 

58( 

>2 

5956 

6050 

6143 

6 

237 

633 

1 

6424 

4 

6518 

6612 

6705 

67$ 

m 

6892 

6986 

7079 

7 

173 

726 

3 

7360 

5 

7453 

7546 

7640 

7733 

7826 

7920 

8013 

8106 

8199 

8293 

6 

8386 

8479 

8572 

86( 

i5 

8759 

8852 

8945 

01 

338 

913 

1 

9224 

OQ17 

9410 

9503 

951/ 

»; 

9689 

9782 

9875 

g 

»;? 

yoi  i 

0060 

0153 

93 

8 

670246 

0339 

0431 

0524 

0617 

0710 

0802 

0895 

0988 

1080 

9 

1173 

1265 

1358 

1451 

1543 

1636 

1728 

1821 

1913 

2005 

470 

2098 

2190 

2283 

2375 

2467 

2560 

2652 

2744  2836 

2929 

1 

3021 

3113 

3205 

3297 

3390 

3482 

3574 

3666  I  3758 

385C 

2 

3942 

4034 

4126 

42 

L8 

4310 

4402 

4494 

4 

386 

467 

,' 

4769 

92 

3 

4861 

4953 

5045 

5137 

5228 

5320 

5412 

5503 

5595 

5687 

4 

5778 

5870 

5962 

60. 

>3 

6145 

6236 

6328 

6 

419 

651 

1 

6602 

5 

6694 

6785 

6876 

69< 

38 

7059 

7151 

7242 

7 

333 

742 

4 

7516 

6 

7607 

7698 

7789 

7881 

7972 

8063 

8154 

8245 

8336 

8427 

7 

8518 

8609 
9519 

8700 
9610 

8791 
9700 

8882 
9791 

8973 

'  9882 

9064 
9973 

9155 

9246 

9337 

91 

0063 

0154 

0245 

9 

680336 

0426 

0517 

0607 

0698 

0789 

0879 

0970 

1060 

1151 

480 

1241 

1332 

1422 

1513 

1603 

1693 

1784 

1874 

1964 

2055 

1 

2145 

2235 

2326 

24 

to 

2506 

i  2596 

2686 

2 

777 

286 

7 

2957 

2 

3047 

3137 

3227 

3317 

3407 

|  3497 

3587 

3077 

3767 

3857 

90 

3 

3947 

4037 

4127 

42 

17 

4307 

4396 

4486 

4 

576 

466 

0 

4756 

4 

4845 

4935 

5025 

5114 

5204 

5294 

5383 

5473 

5563 

5652 

5 

5742 

5831 

5921 

60 

10 

6100 

6189 

6279 

6 

368 

645 

8 

6547 

6 

6636 

6726 

6815 

69( 

)4 

6994 

7083 

7172  7 

261 

735 

1 

7440 

7 

7529 

7618 

7707 

7796 

7886 

7975 

8064  8153 

8242 

8331 

89 

8    8420 
g  i   OQHQ 

8509 
9398 

8598 
9486 

8687 
9575 

8776 
9664 

8865 
9753 

8953 
9841 

9042 

9131 

9220 

0019 

0107 

490 

690196 

0285 

0373 

0462 

0550 

0639 

0728 

0816 

0905 

0993 

1 

1081 

1170 

1258 

1347 

14-35 

1524 

1612 

1700 

1789 

1877 

2 

1965 

2053 

2142 

22; 

JO 

2318 

2406 

2494 

o 

583 

267 

1 

2759 

3 

2847 

2935 

3023 

3111 

3199 

3287 

3375 

3463 

3551 

3639 

88 

4 

3727 

3815 

3903 

39J 

n 

4078 

4166 

4254 

4 

342 

443 

0 

4517 

5 

4605 

4693 

4731 

4# 

38 

4956 

5044 

5131 

5 

319 

530 

7 

5394 

6 

5482 

5569 

5657 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

7 

6356 

6444 

6531 

66 

8 

6706 

6793 

6880 

6 

968 

705 

5 

7142 

8 

7229 

7317 

7404 

745 

)1 

7578 

7665 

7752 

7 

339 

792 

i 

8014 

9 

8100 

8188 

8275 

83( 

32 

8449 

8535 

8622 

8709 

8796 

8883 

87 

PROPORTIONAL  PARTS. 

Diff.   1 

2      3 

4 

5 

6 

7 

8 

9 

98    9.8 

19.6    29.4 

39.2 

49.0 

58.8 

68.6 

78.4 

88.2 

97    9.7 

19.4    29.1 

38.8 

48.5 

58.2 

67.9 

77.6 

87.3 

96    9.6 

19.2    28.8 

38.4 

48.0 

57.6 

67.2 

76.8 

86.4 

95  1  9.5 

19.0    28.5 

38.0 

47.5 

57.0 

66.5 

76.0 

85.5 

94    9.4 

18.8    28.2 

37.6 

47.0 

56.4 

65.8 

75.2 

84.6 

93    9.3 

18.6    27.9 

37.2 

46.5 

55.8 

65.1 

74.4 

92    9.2 

18.4    27.6 

36.8 

46.0 

55.2 

64.4 

73.6 

82^8 

91    9.1 

18.2    27.3 

36.4 

45.5 

54.6 

63.7 

72.8 

81.9 

90    9.0 

18.0    27.0 

36.0 

45.0 

54.0 

63.0 

72.0 

81.0 

89    8.9 

17.8    26.7 

35  6 

44.5 

53.4 

62.3 

71.2 

80.1 

88    8.8 

17.6    26.4 

35.2 

44.0    52.8 

61.6 

70.4 

79.2 

87    8.7   17.4  1  26.1 

34.8 

43.5 

52.2 

60.9 

69.6 

78:3 

86    8.6   17.2  1  25.8 

34.4 

43.0 

51.6 

60.2 

68.8 

77.4 

91 


TABLE    IX. — LOGARITHMS    OF    NUMBERS. 


No.  500  L.  698.]                                   [No.  544  L.  736. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

500 

698970 

9057 

9144 

9231 

9317 

9404 

9491 

9578 

9664 

9751 

9838 

9924 

0011 

0098 

0184 

0271 

0358 

0444 

0531 

0617 

2 

700704 

0790 

0877 

0963 

1050 

1136 

1222 

1309 

1395 

1482 

3 

1568 

1654 

1741 

1827 

1913 

1999 

2086 

2172 

2258 

2344 

4 

2431 

2517 

2603 

2689 

2775 

2861 

2947 

3033 

3119 

3205 

5 

3291 

3377 

3463 

3549 

3635 

3721 

3807 

3893 

3979 

4065 

86 

6 

4151 

4236 

4322 

4408 

4494 

4579 

4665 

4751 

4837 

4922 

7 

5008 

5094 

5179 

5265 

5350 

5436 

5522 

5607 

5693 

5778 

8 

5864 

5949 

6035 

6120 

62C6 

6291 

6376 

6462 

6547 

6632 

9 

6718 

6803 

6888 

6974 

7059 

7144 

7229 

7315 

7400 

7485 

510 

7570 

7655 

7740 

7826 

7911 

7996 

8081 

8166 

8251 

8336 

OK 

1 

8421 

8506 

8591 

8676 

8761 

8846 

8931 

9015 

9100 

9185 

oo 

2 

9270 

9355 

9440 

9524 

9609 

9694 

9779 

9863 

9948 



0033 

3  i  710117 

0202 

0287 

0371 

0456 

0540 

0625 

0710 

0794 

0879 

4    0963 

1048 

1132 

1217 

1301 

1385 

1470 

1554 

1639 

1723 

5    1807 

1892 

1976 

2060 

2144 

2229 

2313 

2397 

2481 

2566 

6 

2650 

2734 

2818 

2902 

2986 

3070 

3154 

3238 

3323 

3407 

04 

7 

3491 

3575 

3659 

3742 

3826 

3910 

3994 

4078 

4162 

4246 

84 

8 

4330 

4414 

4497 

4581 

4665 

4749 

4833 

4916 

5000 

5084 

9 

5167 

5251 

5335 

5418 

5502 

5586 

5669 

5753 

5836 

5920 

520 

6003 

6087 

6170 

6254 

6337 

6421 

6504 

6588 

6671 

6754 

1 

6838 

6921 

1004 

7088 

7171 

7254 

7338 

7421 

7504 

7587 

2 

7671 

7754 

7837 

7920 

8003 

8086 

8169 

8253 

8336 

8419 

3 

8502 

8585 

8668 

8751 

8834 

8917 

9000 

9083 

9165 

9248 

83 

4 

9331 

9414 

9497 

9580 

9663 

9745 

9828 

9911 

9994 

0077 

5 

720159 

0242 

0325 

0407 

0490 

0573 

0655 

0738 

0821 

0903 

6 

0986 

1068 

1151 

1233 

1316 

1398 

1481 

1563 

1646 

1728 

7 

1811 

1893 

1975 

2058 

2140 

2222 

2305 

2387 

2469 

2552 

8 

2634 

2716 

2798 

2881 

2963 

3045 

3127 

3209 

3291 

3374 

9 

3456 

3538 

3620 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

82 

530 

4276 

4358 

4440 

4522 

4604 

4685 

4767 

4849 

49«3l 

5013 

1 

5095 

5176 

5258 

5340 

5422 

5503 

5585 

5667 

5748 

5830 

o 

5912 

5993 

6075 

6156 

6238 

6320 

6401 

6483 

6564 

6646 

3 

6727 

6809 

6890 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

4 

7541 

7623 

7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

5 

8354 

8435 

8516 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

6 

.9165 

9246 

9327 

9408 

9489 

9570 

9651 

9732 

9813 

9893 

81 

7 

9974 

0055 

0136 

0217 

0298 

0378 

0459 

0540 

0621 

0702 

8 

730782 

0863 

0944 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

9 

1589 

1669 

1750 

1830 

1911 

1991 

2072 

2152 

2233 

2313 

540 

2394 

2474 

2555 

2635 

2715 

2796 

2876 

2956 

3037 

3117 

1 

3197 

3278 

3358 

3438 

3518 

3598 

3679 

3759 

3839 

3919 

2 

3999 

4079 

4160 

4240 

4320 

4400 

4480 

4560 

4640 

4720 

Of) 

3 

4800 

4880 

4960 

5040 

5120 

5200 

5279 

5359 

5439 

5519 

ou 

4 

5599 

5679 

5759 

5838 

5918 

5998 

6078 

6157 

6237 

6317 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

678 

9 

87    8.7 

17.4    26  1    34.8 

43  5 

52.2  !  60.9    69  6 

78  3 

86    8.6 

17.2    258    34.4 

43  0 

51  6    60  2    68.8 

77  4 

85    8.5 

17.0    25  5    34.0 

42  5 

51.0    59.5    68.0 

76  5 

84    8.4 

16.8    252    33.6 

42  0 

50.4    58  8    67.2 

75.6 

92 


TABLE   IX. — LOGARITHMS   OF   NUMBERS. 


No.  545  L.  736.] 

LNo.  584  L.  767. 

N. 

0 

1 

2 

8 

4 

5 

i 

7 

8  !  9 

Diff. 

545 

736397 

6476 

6556 

6635 

6715  ||  6795 

~6874~ 

6954 

7034 

7113 

6 

7193 

7272 

7352 

74S 

1 

7511   7590 

7670 

r49 

7829 

7908 

7 

7987 

8067 

8146 

822 

5 

8305   8384 

8463 

8 

543 

8622 

8701 

8 

8781 

8860 

8939 

9018  9097  ||  9177 

9256 

9335 

9414 

9493 

9 

9572 

9051 

9731 

981 

o 

QKRQ    QQI58 

0047 

o 

!•><•. 

none 

O9ft4. 

550 

740363 

0442 

0521  0000 

0078  |  0757 

0836 

Ul/vVJ 

0915 

U*UO 

0994 

1073 

1 

1152 

1230 

1309 

138 

18 

1467  j  1546 

1624 

1 

ro3 

1782 

1860 

o 

1939 

2018  ! 

2096 

217 

5 

2254   2332 

2411 

2 

489 

2568 

2647 

3 

2725 

2804  i 

2882 

29€ 

1 

3039  I  3118 

3196 

3 

275 

3353 

3431 

4    3510 

3588  i 

3007 

3745 

3823  l  3902 

3980 

4058 

4136 

4215 

5    4293 

4371  i 

4449 

452 

8 

4606 

4684 

4762 

4 

340 

4919 

4997 

6 

5075 

5153  ! 

5231 

5309 

5387 

5465 

5543 

5621 

5699 

5777 

78 

7 

-  5855 

5933  ; 

6011 

60S 

9  6167 

6245 

6323 

6 

401 

6479 

6556 

8 

6634 

6712 

6790  i  6868  6945 

'  7023  7101 

7179 

7256 

7334 

9 

7412 

7489 

7507   7045 

7722 

7800  7878 

7955 

8033 

8110 

560 

8188 

8266 

8343 

8421 

8498 

8576  8653 

8731 

8808 

8885 

1 

8963 

9040 

9118 

911 

5  9272 

9350  9427 

9 

504 

9582 

9659 

2 

9736 

QS14 

9891 

99t 

g 

fV\i\ 

0123  0200 

O1 

T-T    AQ^/l 

0431 

3 

750508  0586 

0663  0740  0817 

C894  0971 

1048 

1125 

1202 

4 

1279 

1356 

1433  151 

0  1587 

16U4  1741 

1 

818 

1895 

1972 

5 

2048 

2125 

2202  22' 

•9  2356 

2433  2509 

2 

586 

2663 

2740 

77 

6 

2816 

2893  i 

2970  30- 

7 

3123 

3200  3277 

3353 

3430 

3506 

7 

3583 

3660 

3736  381 

3 

3889 

3966  4042 

4 

119 

4195 

4272 

8 

4348 

4425  I 

4501  4578 

4654 

4730  4807 

4883 

4960 

5036 

9 

5112 

5189 

5205  5341 

5417 

.  5494  5570 

5646 

5722 

5799 

570 

5875 

5951  | 

6027  6103 

6180 

'  6256  6332 

6408 

6484 

6560 

1 

6636 

6712 

6788 

6804 

6940 

7016  7092 

7168 

7244 

7320 

76 

2 

7396 

7472 

7548  76< 

>4 

7700 

7775  7851 

927 

8003 

8079 

3 

8155 

8230 

8306  8& 

V 

8458 

8533  8609 

8 

685 

8761 

8836 

4 

8912 

8988 

9063  j  9139  9214 

9290  9366 

9441 

9517 

9592 

5 

9668 

9743 

9819  98i 

)4   QQ7fl 

0045  0121 

O 

1Q« 

0272 

0347 

6 

760422 

0498 

0573 

0049 

0724 

0799  0875 

0950 

1025 

1101 

7 

1176 

1251 

1326  1402 

1477 

i  1552  1627 

1 

702 

1778 

1853 

8 

1928 

2003 

2078  211 

)3 

2228 

2303  2378 

2 

453 

2529 

2604 

9 

2679 

2754 

2829  2904 

2978 

3053  ,  3128 

3203 

3278 

3353 

<o 

580 

3428 

3503 

3578  3653 

3727 

3802  3877 

3952 

4027 

4101 

1 

4176 

4251 

4326  !  44( 

X) 

4475 

4550  4624 

4 

699 

4774 

4848 

2 

4923 

4998 

5072  51< 

17 

5221 

;  5296  5370 

5 

445 

5520 

5594 

3 

5669 

5743 

5818 

5892 

5966 

6041 

6115 

6190 

6264 

6338 

4 

6413 

6487 

6562 

6& 

JO 

6710 

6785  6859 

6933 

7007 

7082 

PROPORTIONAL  PARTS. 

Diff.   1 

2 

3 

4 

5 

6 

7      8 

9 

83    8.3 

16.6 

24.9 

33.2 

41.5 

49.8 

58.1    66.4 

74.7 

82    8.2 

16.4 

24.6 

32.8 

41.0 

49.2 

57.4    65.6 

73.8 

81    8.1 

16.2 

24.3 

32.4 

40.5 

48.6 

56.7    64.8 

72.9 

80    8.0 

10.0 

24.0 

32.0 

40.0 

48.0 

56.0    64.0 

72.0 

I9  Z-9 

15.8 

23.7 

31.6 

39.5 

47.4 

55.3    63.2 

71.1 

1  8     .8 

15.6 

23.4 

31.2 

39.0 

46.8 

54.6    62.4 

70.2 

77    r'.7 

15.4 

23.1 

30.8 

38.5 

46.2 

53.9    61.6 

69.3 

76    "6 

15.2 

22.8 

30.4 

38.0 

45.6 

53.2    60.8 

68.4 

75    7.5 

15.0 

22.5 

30.0 

37.5 

45.0 

52.5    60.0 

67.5 

74    -.4 

14.8 

22.2 

29.6 

37.0 

44.4 

51.8    59.2 

66.6 

93 


TABLE   IX.— LOGARITHMS   OF   NUMBERS. 


No.  585  L.  767.1                                 [No.  629  L.  799. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

585 

76715G 

7230 

7304 

7379 

7453 

7527  7G01 

7675 

7749 

7823 

6 

7898 

7972 

8046 

8120 

8194 

8268  8:342 

8416 

8490 

8564 

74 

7 

8638 

8712 

•8786 

8860 

8934 

9008  9082 

9156 

9230 

9303 

g 

9377 

9451 

9525 

9599 

9673 

9746  9820 

9894 

9968 

0042 

9 

770115 

0189 

02C3 

0336 

0410 

0484 

0557 

0631 

0705 

0778 

590 

0852 

0926 

0999 

1073 

1146 

1220 

1293 

1367 

1440 

1514 

1 

1587 

1661 

1734 

1808 

1881 

1955 

2028 

2102 

2175 

2248 

2 

2322 

2395 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

3 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

4 

3786 

3860 

3933 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

73 

5 

4517 

4590 

4663 

4736 

4809 

4882 

4955 

5028 

5100 

5173 

6 

5246 

5319 

5392 

5465 

5538 

5610 

5683 

5756 

5829 

5902 

7 

5974 

6047 

6120 

6193 

6265 

6338 

6411 

6483 

6556 

6629 

8 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

9 

7427 

7499 

7572 

7644 

7717 

7789 

7862 

7934 

8006 

8079 

600 

8151 

8224 

8296 

8368 

8441 

8513 

8585 

8658 

8730 

8802 

1 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

2 

9596 

9669 

9741 

9813 

9885 

9957 

0029 

0101 

0173 

0245 

3 

780317 

0389 

0461 

0533 

0605 

0677 

0749 

0821 

0893 

0965 

72 

4 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1540 

1612 

1684 

5 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

6 

2473 

2544 

2616 

2688 

2759 

2831 

2902 

2974 

3046 

3117 

7 

3189 

3260 

3332 

3403 

3475 

3546 

3618 

3689 

3761 

3832 

8 

3904 

3975 

4046 

4118 

4189 

4261 

4332 

4403 

4475 

4546 

9 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5259 

610 

5330 

5401 

5472 

5543 

5615 

5686 

5757 

5828 

5899 

5970 

1 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6538 

6609 

6680 

71 

2 

6751 

6822 

6893 

6964 

7035 

7106 

7177 

7248 

7319 

7390 

3 

7460 

7531 

7602 

7673 

7744 

7815 

7885 

7956 

8027 

8098 

4 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663 

8734 

8804 

5 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

9581 

9651 

9722 

9792 

9863 

9933 

0004 

0074 

0144 

QOJ5 

7 

790285 

0356 

0426 

0496 

0567 

0637 

0707 

0778 

0848 

0918 

8 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

1550 

1620 

9 

1691 

1761 

1831 

1901 

1971 

2041 

2111 

2181 

2252 

2322 

620 

2392 

2462 

2532 

2602 

2672 

2742 

2812 

2882 

2952 

3022 

70 

1 

3092 

3162 

3231 

3301 

3371 

3441 

3511 

3581 

3651 

3721 

2 

3790 

3860 

3930 

4000 

4070 

4139 

4209 

4279 

4349 

4418 

3 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

5045 

5115 

4 

5185 

5254 

5324 

5393 

5463 

5532 

5602 

5672 

5741 

5811 

5 

5880 

5949 

6019 

6088 

6158 

6227 

6297 

6366 

6436 

6505 

6 

6574 

6644 

6713 

6782 

6852 

6921 

6990 

7060 

7129 

7198 

7 

7268 

7337 

7406 

7475 

7545 

7614  1  7683 

7752 

7821 

7890 

8 

7960 

8029 

8098 

8167 

8236 

8305  !  8374 

8443 

8513 

8582 

9 

8651 

8720 

8789 

8858 

8927 

8996 

9065 

9134 

9203 

9272 

69 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

678 

9 

75    7.5 

15.0    22.5    30.0 

37.5 

45.0    52.5    60.0 

67.5 

74    7.4 

14.8    22.2    29.6 

37.0 

44.4    51.8    59.2 

66.6 

73    7.3 

14.6    21.9    29.2 

36.5 

43.8    51.1    58.4 

65.7 

72    7.2 

14.4    21.6    28.8 

36.0 

43.2    50.4    57.6 

64.8 

71    7.1 

14.2    21.3    28.4 

35.5 

42.6    49.7    56.8 

63.9 

70    7.0 

14.0    21.0    28.0 

35.0 

42.0    49.0    56.0 

63.0 

69    6.9 

13.8    20.7    27.6 

34.5 

41.4    48.3    55.2 

62.1 

94 


TABLE    IX. — LOGARITHMS    OF    NUMBERS. 


Ko.  630  L.  799.]                                 [No.  674  L.  829. 

:N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

f630 

799341 

9409 

9478 

9547 

9616 

9685 

9754 

9823 

9892 

9961 

1 

800029 

0098 

0167 

0236 

0305 

0373 

0442 

0511 

0580  j  0648 

2 

0717 

0786 

0854 

0923 

0992 

1061 

1129 

1198 

1266 

1335 

3 

1404 

1472 

1541 

1609 

1678 

1747 

1815 

1884 

1952 

2021 

4 

2089 

2158 

2226 

2295 

2363 

2432 

2500 

2568 

2637 

2705 

5 

2774 

2842 

2910 

2979 

3047 

3116 

3184 

3252 

3321 

a389 

6 

3457 

3525 

3594 

3662 

3730 

3798 

3867 

39&5 

4003 

4071 

7 

4139 

4208 

4276 

4344 

4412 

4480 

4548 

4616 

4685 

4753 

8 

4821 

4889 

4957 

5025 

5093 

5161 

5229 

5297 

5365 

5433 

68 

9 

5501 

5569 

5637 

5705 

5773 

5841 

5908 

5976 

6044 

6112 

640 

806180 

6248 

6316 

6384 

6451 

6519 

6587 

6655 

6723 

6790 

1 

6858 

6926 

6994 

7061 

7129 

7197 

7264 

7332 

7400 

7467 

2 

7535 

7603 

7670 

7738 

7806 

7873 

7941 

8008 

8076 

8143 

3 

8211 

8279 

8346 

8414 

8481 

8549 

8616 

8684 

8751 

8818 

4 

8886 

8953 

9021 

9088 

9156 

9223 

9290 

9358 

9425 

9492 

5 

9560 

9627 

9694 

9762 

9829 

9896 

9964 

0031 

0098 

0165 

6 

810233 

0300 

0367 

0434 

0501 

0569 

0&36 

0703 

0770 

0837 

7 

0904 

0971 

1039 

1106 

1173 

1240 

1307 

1374 

1441 

1508 

67 

8 

1575 

1642 

1709 

1776 

1843 

1910 

1977 

2044 

2111 

2178 

9 

2245 

2312 

2379 

2445 

2512 

2579 

2646 

2713 

2780 

2847 

650 

2913 

2980 

3047 

3114 

3181 

3247 

3314 

3381 

3448 

3514 

1 

3581 

3648 

3714 

3781 

3848 

3914 

3981 

4048 

4114 

4181 

2 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

3 

4913 

4980 

5046 

5113 

5179 

5246 

5312 

5378 

5445 

5511 

4 

5578 

5644 

5711 

5777 

5843 

5910 

5976 

6042 

6109 

6175 

5 

6241 

6308 

6374 

6440 

6506 

6573 

6639 

6705 

6771 

6838 

6 

6904 

6970 

7036 

7102 

7169 

7235 

7301 

7367 

7433 

7499 

n 

7565 

7631 

7698 

7764 

7830 

7896 

7962 

8028 

8094 

8160 

8 

8226 

8292 

8358 

8424 

8490 

8556 

8622 

8688 

8754 

8820 

Cf 

9 

8885 

8951 

9017 

9083 

9149 

9215 

9281 

9346 

9412 

9478 

OD 

660 

9544 

9610 

9676 

9741 

9807 

9873 

9939 

0004 

0070 

0136 

1 

820201 

0267 

0333 

0399 

0464 

0530 

0595 

OG61 

0727 

0792 

2 

0858 

0924 

0989 

1055 

1120 

1186 

1251 

1317 

1382 

1448 

3 

1514 

1579 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

4 

2168 

2233 

2299 

2364 

2430 

2495 

2560 

2626 

2691 

2756 

5 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

6 

3474 

3539 

3605 

3670 

3735 

3800 

3865 

3930 

3996 

4061 

7 

4126 

4191 

4256 

4321 

4386 

4451 

4516 

4581 

4646 

4711 

65 

8 

4776 

4841 

4906 

4971 

5036 

5101 

5166 

5231 

5296 

5361 

9 

5426 

5491 

5556 

5621 

5686 

5751 

5815 

5880 

5945 

6010 

670 

6075 

6140 

6204 

6269 

6334 

6399 

6464 

6528 

6593 

6658 

1 

6723 

6787 

6852 

6917 

6981 

7046 

7111 

7175 

7240 

7305 

2 

7369 

7434 

7499 

7563 

7628 

7692 

7757 

7821 

7886 

7951 

3 

8015 

8080 

8144 

8209 

8273 

8338 

8402 

8467 

8531 

8595 

4 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

PROPORTIONAL  PARTS, 

!Diff    1 

234 

5 

678 

9 

68    68 

13  6    20  4    27  2 

34  0 

40  8    47  G    54  4 

61  2 

67    67 

13  4    20.1    26  8 

33  5 

40  X5    40  '.)    53  6 

CO  3 

66    66 

13.2    19  8    26  4 

33  0 

39  G    46  2    52  8 

59  4 

65    65 

13  0    19  5    26  0 

32.5 

39  0    45  5    52  0 

58  5 

64    6.4 

1£  8    19.2    25  6' 

32  0 

38.4    44  8    51  2 

57.6 

95 


TABLE   IX. — LOGARITHMS   OF   KUMBERS. 


No.  675  L.  829.]                                 [No.  719  L.  857. 

N. 

0 

1 

2 

3  i   4 

6 

6 

7 

S 

9 

Diff. 

! 

675 
6 

829304 
9947 

9368 

9432 

9497 

9561 

9625 

9690 

9754 

9818 

9882 

0011 

0075 

0139 

0204 

0268 

0332 

0396 

0460 

0525 

7 

830589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

8 

1230 

1294 

1358 

1422 

1486 

1550 

1614 

1678 

1742 

1806 

64 

9 

1870 

1934 

1998 

2062 

2126 

2189 

2253 

2317 

2381 

2445 

880 

2509 

2573 

2637 

2700 

2764 

2828 

2892 

2956 

3020 

3083 

1 

3147 

3211 

3275 

3338 

3402 

3466 

3530 

3593 

3657 

3721 

2 

3784 

3848 

3912 

3975 

4039 

4103 

4166 

4230 

4294 

4357 

3 

4421 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993 

4 

5056 

5120 

5183 

5247 

5310 

i  5373 

5437 

5500 

5564 

5627 

5 

5691 

5754 

5817 

5881 

5944 

6007 

6071 

6134 

6197 

6261 

6 

6324 

6387 

6451 

6514 

6577 

6641 

6704 

6767 

6830 

6894 

7 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7462 

7525 

8 

7588 

7652 

7715 

7778 

7841 

7904 

7967 

8030 

8093 

8156 

9 

8219 

8282 

8345 

8408 

8471 

8534 

8597 

8660 

8723 

8786 

63 

690 

8849 

8912 

8975 

9038 

9101 

9164 

9227 

9289 

9352 

9415 

1 

9478 

9541 

9604 

9667 

9729 

9792 

9855 

9918 

9981 



0043 

2 

840106 

0169 

0232 

0294 

0357 

0420 

0482 

0545 

0608 

0671 

3 

0733 

0796 

0859 

0921 

0984 

1046 

1109 

1172 

1234 

1297 

4 

1359 

1422 

1485 

1547 

1610 

1672 

1735 

1797 

1860 

1922 

5 

1985 

2047 

2110 

2172 

2235 

2297 

2360 

2422 

2484 

2547 

6 

2609 

2672 

2734 

2796 

2859 

2921 

2983 

3046 

3108 

3170 

7 

3233 

3295 

3357 

3420 

3482 

3544 

3606 

3669 

3731 

3793 

8 

3855 

3918 

3980 

4042 

4104 

4166 

4229 

4291 

4353 

4415 

9 

4477 

4539 

4601 

4664 

4726 

4788 

4850 

4912 

4974 

5036 

700 

5098 

5160 

5222 

5284 

5346 

5408 

5470 

5532 

5594 

5656 

62 

1 

5718 

5780 

5842 

5904 

5966 

6028 

6090 

6151 

6213 

6275 

2 

6337 

6399 

6461 

6523 

6585 

6646 

6708 

6770 

6832 

6894 

3 

6955 

7017 

7079 

7141 

7202 

7264 

7326 

7388 

7449 

7511 

4 

7573 

7634 

7696 

7758 

7819 

7881 

7943 

8004 

8066 

8128 

5 

8189 

8251 

8312 

8374 

8435 

8497 

8559 

8620 

8682 

8743 

6 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235 

9297 

9358 

7 

9419 

9481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

8 

850033 

0095 

0156 

0217 

0279 

0340 

0401 

0462 

0524 

0585 

9 

0646 

0707 

0769 

0830 

0891 

0952 

1014 

1075 

1136 

1197 

710 

1258 

132.0 

1381 

1442 

1503 

1564 

1625 

1686 

1747 

1809 

1 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

2 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

3 

3090 

3150 

3211 

3272 

3333 

3394 

3455 

3516 

3577 

3637 

4 

3698 

3759 

3820 

3881 

3941 

4002 

4063 

4124 

4185 

4245 

5 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

6 

4913 

4974 

5034 

5095 

5156 

5216 

5277 

5337 

5398 

5459 

7 

5519 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6003 

6064 

8 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6548 

6608 

6668 

9 

6729 

6789 

6850 

6910 

6970 

7031 

7091 

7152 

7212 

7272 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

678 

9 

65    6.5 

13.0    19.5    26.0 

32.5 

39.0    45.5    52.0 

58.5 

64    6.4 

12.8    19.2    25.6 

32.0 

38.4    44.8    51.2 

57.6 

63    6.3 

12.6    18.9    25.2 

31.5 

37.8    44.1    50.4 

56.7 

62  '   6.2 

12.4    18.6    24.8 

31.0 

37.2    43.4    49.6 

55  8 

61    6.1 

12.2    18.3    24.4 

30.5 

36.6    42.7    48.8 

54.9 

60    6.0 

12.0    18.0    24.0 

30.0 

36.0    42.0    48.0 

54.0 

96 


TABLE   IX. — LOGARITHMS   OF   NUMBERS. 


No.  720  L.  857.]                                  [No.  76*  L.  883. 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Diff. 

720 

857332 

7393 

7453 

7513 

7574 

7634 

7694 

7755 

7815 

7875 

1 

7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

2 

8537 

8597 

8657 

8718 

8778 

i  8838 

8898 

8958 

9018 

9078 

3 

9138 

9198 

9258 

9318 

9379 

;  9439 

9499 

9559 

9619 

9679 

60 

Q7QQ 

Q7QQ 

9859 

9918 

9978 

4 

y<oy 

y<yy 

0038 

0098 

0158 

0218 

0278 

5 

860338 

0398 

0458 

0518 

0578 

0637 

0697 

0757 

0817 

0877 

6 

0937 

0996 

1056 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

7 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

1952 

2012 

2072 

8 

2131 

2191 

2251 

2310 

2370 

2430 

2489 

2549 

2608 

2668 

9 

2728 

2787 

2847 

2906 

2966 

3025 

3085 

3114 

3204 

3263 

730 

3323 

3382 

3442 

3501 

3561 

3620 

3680 

3739 

3799 

3858 

1 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

2 

4511 

4570 

4630 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

3 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

5519 

5578 

5637 

4 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

5 

6287 

6346 

6405 

6465 

6524 

6583 

C642 

6701 

6760 

6819 

6 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

59 

7 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

8 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

8468 

8527 

8586 

9 

8644 

8703 

8762 

8821 

8879 

8938 

8997 

9056 

9114 

91  <  3 

740 

9232 

9818 

9290 
9877 

9349 

9935 

9408 
9994 

9466 

9525 

9584 

9642 

9701 

9760 

0053 

0111 

0170 

0228 

0287 

0345 

2 

870404 

0462 

0521 

0579 

0638 

0696 

0755 

0813 

0872 

0930 

3 

0989 

1047 

1100 

1164 

1223 

1281 

1339 

1398 

1456 

1515 

4 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2008 

5 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

6 

2739 

2797 

2855 

2913 

2972 

3030 

3088 

8146 

3204 

3262 

7 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

8 

3902 

3960 

4018 

4076 

4134 

1  4192 

4250 

4308 

4366 

4424 

58 

9 

4482 

4540 

4598 

4656 

4714 

4772 

4830 

4888 

4945 

5003 

750 

5061 

5119 

5177 

5235 

5293 

5351 

5409 

5466 

5524 

5582 

1 

5640 

5698 

5756 

5813 

5871 

|  5929 

5987 

6045 

6102 

6160 

2 

6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

3 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

4 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

7-774 

7832 

7889 

5 

7947 

8004 

8062 

8119 

8177 

8234 

8292 

8349 

8407 

8464 

6 

8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

7 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9555 

9612 

g 

9669 

9726 

9784 

9841 

9898 

9956 

0013 

0070 

0127 

0185 

9 

880242 

0299 

0356 

0413 

0471 

0528 

0585 

0642 

0699 

0756 

760 

0814 

0871 

0928 

0985 

1042 

1099 

1156 

1213 

1271 

1328 

1 

1385 

1442 

1499 

1556 

1613 

1670 

1727 

1784 

1841 

1898 

2 

1955 

2012 

2069 

2126 

2183 

2240 

2297 

2354 

2411 

2468 

57 

3 

2525 

2581 

2638 

2695 

2752 

2809 

2866 

2923 

2980 

3037 

4 

3093 

3150 

3207 

3264 

3321 

3377 

3434 

3491 

3548 

3605 

1 

PROPORTIONAL  PARTS. 

Diff.   1 

2 

3      4 

5 

678 

9 

59    5.9 

11.8 

17.7    23.6 

29.5 

35.4    41.3    47.2 

53.1 

58    5.8 

11.6 

17.4    23.2 

29.0 

34.8    40.6    46.4 

52.2 

57    5.7 

11.4 

17.1    22.8 

28.5 

34.2    39.9    45.6 

51.3 

56    5.6 

11.2 

16.8    22.4 

28.0    33.6    39.2    44.8 

50.4 

97 


TABLE    IX. — LOGARITHMS    OF    NUMBERS. 


No.  765  L.  883.]                                  [No.  809  L.  908. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

765 

883661 

3718 

3775 

3832 

3888 

3945 

4002 

4059 

4115 

4172 

6 

4229 

4285 

4342 

4399 

4455 

4512 

4569 

4625 

4682 

4739 

7 

4?95 

4852 

4909 

4965 

5022 

5078 

5135 

5192 

5248 

5305 

8 

5361 

5418 

5474 

5531 

5587 

5644 

5700 

5757 

5813 

5870 

9 

5926 

5983 

6039 

6096 

6152 

6209 

6265 

6321 

6378 

6434 

770 

6491 

6547 

6604 

6660 

6716 

6773 

6829 

6885 

6942 

6998 

1 

7054 

7111 

7167 

7223 

7280 

7336 

7392 

7449 

7505 

7561 

2 

7617 

7674 

7730 

7786 

7842 

7898 

7955 

8011 

8067 

8123 

3 

8179 

8236 

8292 

8348 

8404 

8460 

8516 

8573 

8629 

8685 

4 

8741 

8797 

8853 

8909 

8965 

9021 

9077 

9134 

9190 

9246 

5 

Q 

9302 

9862 

9358 
9918 

9414 
9974 

9470 

9526 

9582 

90138 

9694 

9750 

9806 

56 

0030 

0086 

0141 

0197 

0253 

0309 

0365 

7 

890421 

0477 

0533 

0589 

0645 

0700 

0756 

0812 

0868 

0924 

8 

0980 

1035 

1091 

1147 

1203 

1259 

1314 

137'0 

1426 

1482 

9 

1537 

1593 

1649 

1705 

1760 

1816 

1872 

1928 

1983 

2039 

780 

2095 

2150 

2206 

2262 

2317 

2373 

2429 

2484 

2540 

2595 

1 

2651 

2707 

2762 

2818 

287'3 

2929 

2985 

3040 

3096 

3151 

2 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

3 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

4 

4316 

4371 

4427 

4482 

4538 

4593 

4648 

4704 

4759 

4814 

5 

4870 

4925 

4980 

5036 

5091 

5146 

5201 

5257 

5312 

5367 

6 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

5864 

5920 

7 

5975 

6030 

6085 

6140 

6195 

6251 

6306 

6361 

6416 

6471 

8 

6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912 

6967 

7022 

9 

7077 

7132 

7187 

7242 

7297 

7352 

7407 

7462 

7517 

7572 

55 

790 

7627 

7682 

7737 

7792 

7847 

7902 

7957 

8012 

8067 

8122 

1 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

8670 

2 

8725 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

3 

A 

9273 
9821 

9328 
9875 

9383 
9930 

9437 
9985 

9492 

9547 

9602 

9656 

9711 

9766 

0039 

0094 

0149 

0203 

0258 

0312 

5 

900367 

0422 

0476 

~053T 

0586 

0640 

0695 

0749 

0804 

0859 

6 

0913 

0968. 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

7 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

8 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

9 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

800 

3090 

3144 

3199 

3253 

3307 

3361 

3416 

3470 

3524 

3578 

1 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

2 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

3 

4716 

4770 

4824 

4878 

4932 

4986 

5040 

5094 

5148 

5202 

54 

4 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

5 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

6 

6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

7 

6874 

6927 

6981 

7035 

7089 

7143 

7196 

7250 

7304 

7358 

8 

7411 

7465 

7519 

7573 

7626 

7680 

7734 

7787 

7841 

7895 

9 

7949 

8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

678 

9 

57    5.7 

11.4    17.1    22.8 

28.5 

34.2    39.9    45.6 

51.3 

56    5.6 

11.2    16.8    22.4 

28.0 

33.6    39.2    44.8 

50.4 

55    5.5 

11.0    16.5    22.0 

27.5 

33.0    38.5    44.0 

49.5 

54    5.4 

10.8    16.2    21.6 

27.0 

32.4    37.8    43.2 

48.6 

98 


TABLE    IX. — LOGARITHMS   OF   NUMBERS. 


No.  810  L.  908.]                                  [No.  854  L.  931. 

N. 

0 

1 

2 

1 

«  1  6 

6 

7 

8 

9 

Diff. 

810 

908485 

8539 

8592 

8646 

8699 

8753 

8807 

8860 

8914 

8967 

1 

9021 

9074 

9128 

9181 

9235 

9289 

9342 

9396 

9449  |  9503 

2 

9556 

9610 

9603 

9716 

9770 

9823 

9877 

9930 

9981 

nnQT 

3 

910091 

0144 

0197 

0251 

0304 

0358 

0411 

0404 

0518 

0571 

4 

0624 

0678 

0731 

0784 

0838 

0891 

0944 

0998 

1051 

1104 

5 

1158 

1211 

1264 

1317 

1371 

1424 

1477 

1530 

1584 

1637 

6 

1690 

1743 

1797 

1850 

1903 

1956 

2009 

2063 

2116 

2169 

7 

2222 

2275 

21328 

2381 

2435 

2488 

2541 

2594 

2647 

2700 

8 

2753 

2806 

2859 

2913 

2966 

3ul9 

3072 

3125 

3178 

3231 

9 

3284 

3337 

3390 

3443 

3496 

3549 

3002 

3655 

3708 

3761 

53 

820 

3814 

3867 

3920 

3973 

4026 

4079 

4132 

4184 

4237 

4290 

1 

4:343 

4396 

4449 

4502 

4555 

4608 

4660 

4713 

4766 

4819 

2 

4872 

4925 

4977 

5030 

5083 

5136 

5180 

5241   5294 

5347 

3 

5400 

5453 

5505 

5558 

5611 

5664 

5716 

5709 

5822 

5875 

4 

5927 

5980 

6033 

6085 

6138 

6191 

6243 

6296 

6349 

6401 

5 

6454 

6507 

6559 

6612 

6664 

6717 

6770 

(822 

6875 

6927 

6 

6980 

7033 

7085 

7138 

7190 

7243 

7295 

7348 

7400 

7453 

7 

7506 

7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

8 

8030 

8083 

8135 

8188 

8240 

8293 

8345 

8397 

8450 

8502 

9 

8555 

8607 

8659 

8712 

8764 

8816 

8869 

8921 

8973 

9026 

830 

9078 

9130 

9183 

9235 

9287 

9340 

9392 

9444 

9496 

9549 

1    9601 

9653 

<i"n<  i 

9758 

9810 

OftflO 

9914 

9907 

!  nmn 

0071 

2 

920123 

0176 

0228 

0280 

0332 

0384 

0436 

0489 

0541 

0593 

3 

0645 

0697 

0749 

0801 

0853 

0906 

0958 

1010 

1062 

1114 

52 

4 

1166 

1218 

1270 

1322 

1374 

1426 

1478 

1530 

1582 

1634 

5 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2050 

2102 

2154 

6 

2206 

2258 

2310 

2362 

2414 

2466 

2518 

2570 

2622 

2674 

7 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

8 

3244 

3296 

3348 

3399 

3451 

3503 

3555 

3607 

3658 

3710 

9 

3762 

3814 

3865 

3917 

3969 

4021 

4072 

4124 

4176 

4228 

840 

4279 

4331 

4383 

4434 

4486 

4538 

4589 

4641 

4693 

4744 

1 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

5261 

2 

5312 

5364 

5415 

5467 

5518 

5570 

5621 

5673 

5725 

5776 

3 

5828 

5879 

5931 

5982 

6034 

6085 

6137 

6188 

6240 

6291 

4 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

6754 

6805 

5 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

6 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

7 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

8 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8754 

8805 

8857 

9 

8908 

8959 

9010 

9061 

9112 

9163 

9215 

9266 

9317 

9368 

850 
1 

9419 
9930 

9470 

9981 

9521 

9572 

9623 

9674 

9725 

9776 

9827 

9879 

51 

0032 

0083 

0134 

0185 

0236 

0287 

0338 

0389 

2 

930440 

0491 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

3 

0949 

1000 

1051 

1102 

1153 

1204 

1254 

1305 

1356 

1407 

4 

1458 

1509 

1560 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

PROPORTIONAL,  PARTS. 

Diff.   1 

234 

5 

6      7 

8 

9 

53    5.3 

10.6    15.9    21.2 

26.5 

31.8    37.1 

42.4 

47.7 

52    5.2 

10.4    15.6    20.8 

26.0 

31.2    36.4 

41.6 

46.8 

51    5.1 

10.2    15.3    20.4 

25.5 

30.6    35.7 

40.8 

45.9 

50    5.0 

10.0    15.0    20.0 

25.0 

30.0    35.0 

40.0 

45.0 

99 


TABLE    IX. — LOGARITHMS   OF    NUMBERS. 


No.  855  L.  931.]                                  [No.  899  L.  954. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

1 

355 

931966 

2017 

2068 

2118 

2169 

2220  2271  !  2322 

2372 

2423 

6 

2474 

2524 

2575 

2626 

2677 

2727 

2778  i  2829 

2879 

2930 

7 

2981 

3031 

3082 

3133 

3183 

3234  3285  3335 

3386 

3437 

8 

3487 

3538 

3589 

3639 

3690 

!  3740 

3791  j  3841 

3892 

3943 

9 

3993 

4044 

4094 

4145 

4195 

4246 

4296 

!  434? 

4397 

4448 

860 

4498 

4549 

4599 

4650 

4700 

4751 

4801 

4852 

4902 

4953 

1 

5003 

5054 

5104 

5154 

5205 

5255 

5306 

5356 

5406 

5457 

2 

5507 

5558 

5608 

5658 

5709 

5759 

5809 

5800 

5910 

5960 

3 

6011 

6061 

6111 

6162 

6212 

6262 

6313 

6363 

6413 

6463 

4 

6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

5 

7016 

7066 

7116 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

6 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

7 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

&170 

50 

8 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

9 

9020 

9070 

9120 

9170 

9220 

9270 

9320 

9369 

9419 

9469 

870 

9519 

9569 

9619 

9669 

9719 

9769 

9819 

9869 

9918 

9968 

1 

940018 

0068 

0118 

0168 

0218 

0267 

0317 

0367 

0417 

0467 

2 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865 

0915 

0964 

3 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

4 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859 

1909 

1958 

5 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2455 

6 

2504 

2t>54 

2603 

2653 

2702 

2752 

2801 

2851 

2901 

2950 

7 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

8 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

9 

3989 

4038 

4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433 

880 

4483 

4532 

4581 

4631 

4680 

4729 

4779 

4828 

4877 

4927 

1 

4976 

5025 

5074 

5124 

5173 

5222 

5272 

5321 

5370 

5419 

2    5469 

5518 

5567 

5616 

5665 

5715 

5764 

5813 

5862 

5912 

3 

5961 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6354 

6403 

4 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

5 

6943 

6992 

7041 

7090 

7139 

7189 

7238 

7287 

7336 

7385 

6 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

7777 

7826 

7875 

49 

7 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

8 

8413 

8462 

8511 

8560 

8608 

8657 

8706 

8755 

8804 

8853 

9 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

890 

9390 

9439 

9488 

9536 

9585 

9634 

9683 

9731 

9780 

9829 

1 

9878 

9926 

9975 

0024 

0073 

0121 

0170 

0219 

0267 

0316 

2 

950365 

0414 

0462 

0511 

0560 

0608 

0657 

0706 

0754 

0803 

3 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

1289 

4 

1338 

1386 

1435 

1483 

1532 

1580 

1629 

1677 

1726 

1775 

5 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

6 

2308 

2356 

2405 

2453 

2502 

2550 

2599 

2647 

2696 

2744 

7 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

8 

3276 

3325 

3373 

3421 

3470 

3518 

3566 

3615 

3663 

3711 

9 

3760 

3808 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

PROPORTIONAL  PARTS. 

Diff. 

1 

2 

3      4 

5 

678 

9 

51 
50 
49 

5.1 
5.0 
4.9 

10.2 
10.0 
9.8 

15.3    20.4 
15.0    20.0 
14.7    19.6 

25.5 
25.0 
24.5 

30.6    35.7    40.8 
30.0    35.0    40.0 
29.4    34.3    39  2 

45.9 
45.0 
44.1 

48 

4.8 

9.6 

14.4    19.2 

24.0 

28.8    33.6    38.4 

43.'2 

100 


TABLE    IX. — LOGARITHMS   OF   LUMBERS. 


No  900  L.  954.1                                  [No.  944  L.  975. 

N. 

0 

1 

2 

3 

4 

5 

6 

1 

8 

9 

Diff. 

900 

954243 

4291 

4339 

4387 

4435 

4484 

4532 

4580 

4628  4677 

1 

4725 

4773 

4821 

4869 

4918 

4966 

5014 

5062 

5110 

5158 

2 

5207 

5255 

5:303 

51351 

5399 

5447 

5495 

5543 

5592 

5640 

3 

5688 

5736 

5784 

5832 

5880 

5928 

5976 

6024 

6072 

6120 

4 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

6553 

6601 

5 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

48 

6 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7512 

7559 

7 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

8 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

9 

8564 

8<>12 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

910 

9041 

9089 

9137 

9185 

9232 

9280 

9328 

9375 

9423 

9471 

1 
2 

9518 
9995 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

0042 

0090 

0138  |  m8r; 

0233 

0280 

0328 

0376 

0423 

3 

960471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

4 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

5 

1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

6 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

7 

2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

8 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

9 

3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

920 

3788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

1 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

2 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

5108 

5155 

3 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

5578 

5625 

4 

5672 

5719 

5766 

5813 

5860 

5907 

5954 

6001 

6048 

6095 

«r 

5 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

6 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7033 

7 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7454 

7501 

8 

7548 

7595 

7642 

7688 

7735 

7782 

7829 

7875 

7922 

7969 

9 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

930 

8483 

8530 

R576 

8623 

8670 

8716 

8763 

8810 

8856 

8903 

1 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

2 
3 

9416 

9882 

9463 

9928 

9509 
9975 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

0021 

0068 

0114 

0161 

0207 

0254 

0300 

4 

970347 

0393 

0440 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

5 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

6 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

7 

1740 

1786 

1832 

1879 

1925 

1971 

2018 

2064 

2110 

2157 

8 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

9 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

940 

3128 

3174 

3220 

3266 

a313 

3359 

3405 

3451 

3497 

3543 

1 

3590 

3636 

3682 

3728 

3774 

3820 

3866 

3913 

3959 

4005 

2 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

3 

4512 

4558  !  4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

4 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

5340 

5386 

46 

PROPORTIONAL  PARTS. 

Diff.   1 

234 

5 

678 

9 

47    4.7 

9.4    14.1    18.8 

23.5 

28.2    32.9    37.6 

42.3 

46    4.6 

9.2    13.8    18.4 

23.0 

27.6    32.2    36.8 

41.4 

101 


TABLE    IX. — LOGAKITHMS    OF    NUMBERS. 


No.  945  L.  975.]                                 [No.  989  L.  995. 

N. 

0 

1 

? 

a 

4 

5 

6 

7 

8 

9 

Diff. 

945 

975432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

6 

5891 

£937 

5983 

6029 

6075 

6121 

6167 

6212 

6258 

6304 

7 

6350 

6396 

6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

8 

6808 

6854 

6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

9 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

950 

7724 

7769 

7815 

7861 

7906 

7952 

7998 

8043 

8089 

8135 

1 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

2 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

9047 

3 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

4 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

5 

980003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

6 

C458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

7 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

8 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

9 

1819 

1864 

1909 

1954 

2000 

2045 

2090 

2135 

2181 

2226 

960 

2271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

1 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

2 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

3 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

4 

4077 

4122 

4167 

4212 

4257 

4302 

4347 

4392 

4437 

4482 

5 

4527 

4572 

4617 

4662 

47'07 

4752 

4797 

4842 

4887 

4932 

45 

6 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5382 

7 

5426 

5471 

5516 

5561 

5606 

5651 

5696 

5741 

5786 

5830 

8 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

9 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

970 

6772 

6817 

6861 

6906 

6951 

6996 

7040 

7085 

7130 

7175 

1 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

2 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

3 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

4 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

5 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

6 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

9895 

9939 

9983 

0028 

0072 

0117 

0161 

0206 

0250 

Ot-J94 

8 

990339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

9 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

980 

1226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

1 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

2 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

3 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

4 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

5 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

6 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

7 

4317 

4361 

4405 

4449 

4403 

4537 

4581 

4625 

4669 

4713 

44 

8 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

5065 

5108 

5152 

9 

5196 

5240 

5284 

5328 

5372 

5416 

5460 

5504 

5547 

5591 

PROPORTIONAL  FARTS. 

Diff 

1 

2 

3      4 

5 

678 

9 

46 

4.6 

9.2 

13.8    18.4 

23.0 

27.6    32.2    36.8 

41.4 

45 

4.5 

9.0 

13.5    18.0 

22.5 

27.0    31.5    36.0 

40.5 

44 

4.4 

8.8 

13.2    17.6 

22.0 

26.4    30.8    35.2 

39  6 

43 

4.3 

8.6 

12.9    17.2 

21.5 

25.8    30.1    34.4 

38.7 

102 


TABLE    IX. — LOGARITHMS   OF   NUMBERS. 


No.  990  L.  995.] 

[No.  999  L.  999. 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

990 

995635 

5679 

5723 

5767 

5811 

5854 

5898 

5942 

5986 

6030 

1 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

0380 

6424 

6468 

44 

2 

6512 

6555 

6599 

6043 

6687 

6731 

6774 

6818 

6862 

6906 

3 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

729S 

7343 

4 

7386 

7430 

7474 

7517 

7561 

7605 

7648 

7692 

773e 

7779 

5 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

817$ 

8216 

6 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8504 

860£ 

;     8652 

7 

8695 

8739 

8782 

8826 

8869 

8913 

8956 

9000 

904? 

1     9087 

8 

9131 

9174 

9218 

9201 

9305 

9348 

9392 

9435 

9471 

>     9522 

9 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

987C 

9911 

J     9957 

43 

LOGARITHMS  OF  NUMBERS 

FROM  1  TO  100. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

0.000000 

21  1     .322219 

41 

1. 

612784 

61 

1.785330 

81 

.908485 

2 

0.301030 

22        .342423 

42 

1.623249 

62 

1.792392 

82 

.913814 

3 

0.477121 

23 

.361728 

43 

1. 

6334$ 

* 

63 

1.71 

)9341 

83 

.919078 

4 

0.602060 

24 

.380211 

44 

1. 

64345.' 

5 

64 

1.8( 

X5180 

84 

.924279 

5 

0.698970 

25 

.397940 

45 

1. 

653213 

65 

1.812913 

85 

.929419 

6 

0.778151 

26 

.414973 

46 

1. 

662758 

66 

1.819544 

86 

.934498 

7 

0.845098 

27 

.431364 

47 

1. 

6720& 

I 

67 

.8$ 

>6075 

87 

.939519 

8 

0.903090 

28 

.447158 

48 

1. 

681241 

68 

.832509 

as 

.944483 

9 

0.954243 

29 

.462398 

49 

1. 

6901  9( 

> 

69 

.« 

58849 

89 

.949390 

10 

1.000000 

30 

.477121 

50 

1. 

69897C 

) 

70 

.845098 

90 

.954243 

11 

1.041393 

31 

.491362 

51 

1. 

707570 

71 

.851258 

91 

.959041 

12 

1.079181 

32 

.505150 

52 

1. 

71600; 

J 

72 

.« 

>7332 

92 

.963788 

13 

1.113943 

33 

.518514 

53 

1. 

72427( 

1 

73 

.« 

>3323 

93 

.968483 

14 

1  .  146128 

34 

.531479 

54 

1. 

732394 

74 

.869232 

94 

.973128 

15 

1  176091 

35 

.544068 

55 

1. 

740363 

75 

1.875061 

95 

.977724 

16 

1.204120 

36 

.556303 

56 

1 

748188 

76 

1.880814 

96 

.982271 

17 

1.230449 

37 

.568202 

57 

1. 

75587E 

77 

1.8* 

<6491 

97 

.986772 

18 

1.255273 

38 

.579784 

58 

1. 

763428 

78 

1.892095 

98 

.991226 

19 

1.278754 

39 

.591065  ! 

59 

1. 

77085S 

79 

l.« 

17627 

99 

.995635 

20 

1.301030 

40 

.602060 

60 

1.778151  j 

80 

1.903090 

100     2.000000 

Value 

at  0°. 

Sign 
in  1st 
Quad. 

Valm 
at  90° 

>     Sign 
1    in2d 
Quad 

Value 
at 

.     180°. 

Sign     ' 
in  3d 
Quad. 

Value 
at 
270° 

Sign 
in  4th 
Quad. 

Value 
at 
360°. 

Sin    .. 

Q 

R 

_}_ 

0 

R 

Tan  .  .  . 

o 

QO 

o 

4. 

00 

Q 

Sec 

R 

OO 

5 

R 

Versin  

O 

__ 

R 

4- 

2R 

4- 

R 

o 

Cos  

R 



o 

R 

o 

R 

Cot 

00 

__ 

o 

oc 

i 

Q 

Cosec  

GO 

-- 

R 

4- 

CO 

T^ 

R 

- 

00 

R  signifies  equal  to  rad;  oo  signifies  infinite  ;  O  signifies  evanescent. 

103 


TABLE    X. — LOGARITHMIC    SINES, 


" 

/ 

Sine. 

g-l 

Tang. 

Cotang. 

q  +  l 

Dl" 

Cosine. 

/ 

4.685 

15.314 

0 

0 

[nf  .  neg. 

5751 

575 

Inf.  neg. 

Inf.  pos. 

425 

ten 

60 

60 

1 

6.463726 

575 

575 

6.463726 

13.536274 

425 

ten 

59 

120 

2 

.764756 

575 

575 

.764756 

.235244 

425 

ten 

58 

180 

3 

6.940847 

575 

575 

6.940847 

13.059153 

425 

ten 

57 

240 

4 

7.065786 

575 

575 

7.065786 

12.934214 

425 

ten 

56 

300 

5 

.162696 

575 

575 

.162696 

.837304 

425 

ten 

55 

360 

6 

.241877 

575 

575 

.241878 

.758122 

425 

.02 

9.999999 

54 

420 

7 

.308824 

575 

575 

.308825 

.691175 

425 

.00 

.999999 

53 

480 

8 

.366816 

574 

576 

.366817 

.633183 

424 

.00 

.999999 

52 

540 

9 

.417968 

574 

576 

.417970 

.582030 

424 

.00 

.999999 

51 

GOO 

10 

.463726 

574 

576 

.463727 

.536273 

424 

.02 

.999998 

50 

660 

11 

7.505118 

574 

576 

7.505120 

12.494880 

424 

.00 

9.999998 

49 

720 

12 

.542906 

574 

577 

.542909 

.457091 

423 

.02 

.999997 

48 

780 

13 

.577668 

574 

577 

.577672 

.422328 

423 

.00 

.999997 

47 

840 

14 

.609853 

574 

577 

.609857 

.390143 

423 

.02 

.999996 

46 

900 

15 

.639816 

573 

578 

.639820 

.360180 

422 

.00 

.999996 

45 

960 

16 

.667845 

573 

578 

.667849 

.332151 

422 

.02 

.999995 

44 

1020 

17 

.694173 

573 

578 

.694179 

.305821 

422 

.00 

.999995 

43 

1080 

18 

.718997 

573 

579 

.719003 

.280997 

421 

.02 

.999994 

42 

1140 

19 

.742478 

573 

579 

.742484 

.257516 

421 

.02 

.999993 

41 

1200 

20 

.764754 

572 

!580 

.764761 

.235239 

420 

.00 

.999993 

40 

1260 

21 

7.785943 

572 

580 

7.785951 

12.214049 

420 

.02 

9.999992 

39 

1320 

22 

.806146 

572 

1581 

.806155 

.193845 

419 

.02 

.999991 

38 

1380 

23 

.825451 

572 

581 

.825460 

.174540 

419 

.02 

.999990 

37 

1440 

24 

.843934 

571 

1582 

.843944 

.156056 

418 

.02 

.999989 

36 

1500 

25 

.861662 

571 

583 

.861674 

.138326 

417 

.00 

.999989 

35 

1560 

26 

.878695 

571 

583 

.878708 

.121292 

417 

.02 

.999988 

34 

1620 

27 

.895085 

570 

[584 

.895099 

.104901 

416 

.02 

.999987 

33 

1680 

28 

.910879 

570 

1584 

.910894 

.089106 

416 

.02 

.999986 

32 

1740 

29 

.926119 

570 

585 

.926134 

.073866 

415 

.02 

.999985 

31 

1800 

30 

.940842 

569 

586 

.940858 

.059142 

414 

.03 

.999983 

30 

1860 

31 

7.955082 

569 

'587 

7.955100 

12.044900 

413 

.02 

9.999982 

29 

1920 

32 

.968870 

569 

587 

.968889 

.031111 

413 

.02 

.999981 

28 

1980 

33 

.982233 

568 

588 

.982253 

.017747 

412 

.02 

.999980 

27 

2040 

34 

7.995198 

568 

|589 

7.995219 

12.004781 

411 

.02 

.999979 

26 

2100 

35 

8.007787 

567 

590 

8.007809 

11.992191 

410 

.03 

.999977 

25 

2160 

36 

.020021 

567 

591 

.020044 

.979956 

409 

.02 

.999976 

24 

2220 

37 

.031919 

566 

592 

.031945 

.968055 

408 

.02 

.999975 

23 

2280 

38 

.043501 

566 

593 

.043527 

.956473 

407 

.03 

.999973 

22 

2340 

39 

.054781 

566 

!593 

.054809 

.945191 

407 

.02 

.999972 

21 

2400 

40 

.065776 

565 

594 

.065806 

.934194 

406 

.03 

.999971 

20 

2460 

41 

8.076500 

565 

595 

8.076531 

11.923469 

405 

.03 

9.999969 

19 

2520 

42 

.086965 

564 

596 

.086997 

.913003 

404 

.02 

.999968 

18 

2580 

43 

.097183 

564 

598 

.097217 

.902783 

402 

.03 

.999966 

17 

2640 

44 

.107167 

563 

599 

.107203 

.892797 

401 

.03 

.999964 

16 

2700 

45 

.116926 

562 

600 

.116963 

.883037 

400 

.02 

.999963 

15 

2760 

46 

.126471 

562 

601 

.126510 

.873490 

399 

,03 

.999961 

14 

2820 

47 

.135810 

561 

602 

.135851 

.864149 

398 

.03 

.999959 

13 

2880 

48 

.144953 

561 

603 

.144996 

.855004 

397 

.02 

.999958 

12 

2940 

49 

.153907 

560 

604 

.153952 

.846048 

396 

.03 

.999956 

11 

3000 

50 

.162681 

560 

605 

.162727 

.837273 

395 

.03 

.999954 

10 

3060 

51 

8.171280 

559 

607 

8.171328 

11.828672 

393 

.03 

9.999952 

9 

3120 

52 

.179713 

558 

608 

.179763 

.820237 

392 

.03 

.999950 

8 

3180 

53 

.187985 

558 

609 

.188036 

.811964 

391 

.03 

.999948 

7 

3240 

54 

.196102 

557 

1  611 

.196156 

.803844 

389 

.03 

rvq 

.999946 

6 

3300 

55 

.204070 

556 

612 

.204126 

.795874 

388 

,U3 

.999944 

5 

3360 

56 

.211895 

556 

613 

.211953 

.788047 

387 

.03 

.999942 

4 

8420 

57 

.219581 

555 

615 

.219641 

.780359 

385 

.03 

.999940 

3 

3480 

58 

.227134 

554 

616 

.227195 

.772805 

384 

.03 

.999938 

2 

3540 

59 

.234557 

554 

618 

.234621 

.765379 

382 

.03 

.999936 

1 

3600 

60 

8.241855 

553 

1619 

8.241921 

11.758079 

381 

.03 

9.999934 

0 

4.065 

15.314 

// 

/ 

Cosine. 

q-l 

Cotang. 

Tang. 

g  +  Z 

Dl" 

Sine. 

/ 

90° 


104 


TABLE    X. — LOGARITHMIC    SINES, 


If 

/ 

Sine. 

q-l 

Tang. 

Cotang. 

q  +  l 

Dl" 

Cosine. 

/ 

4.685 

15.314 

3600 

C 

8.241855  553  1619 

8.241921 

11.758079 

381  ' 

9.999934 

60 

3660 

1 

.249033 

552 

620 

.249102 

.750898 

380 

.0^ 

.999932 

59 

3720 

8 

.256094 

551 

622 

.256165 

.743835 

378 

•!5   .999929 

58 

3780 

3 

.263042 

551 

623 

.263115 

.736885 

377 

•}5   .999927 

57 

3840 

4 

.269881 

550  , 

625 

.269956 

.730044 

375 

•JJJ   .999925 

56 

3900 

5 

.276614 

549 

627 

.276691 

.723309 

373 

.uo 

riq 

.999922 

55 

3960 

6 

.283243 

548 

628 

.283323 

.716677 

372 

.Uij 
f\n 

.999920 

54 

4020 

7 

.289773 

547 

630 

.289856 

.710144 

370 

.Uo 

.999918 

53 

4080 

8 

.296207 

546 

632 

.296292 

.703708 

368 

.05 

AQ 

.999915 

52 

4140 

9 

.302546 

546 

633 

.302634 

.697366 

367 

.Uo 

.999913 

51 

4200 

10 

.308794 

545 

635 

.308884 

.691116 

365 

.05 

.999910 

50 

4260 

11 

8.314954 

544 

637 

8.315046 

11.684954 

363 

.05 

9.999907 

49 

4320 

12 

.321027 

543 

638 

.321122 

.678878 

30°   >uo 

.999905 

48 

4380 

13 

.327016 

542 

640 

.327114 

.672886 

300   •}* 

.999902 

47 

4440 

14 

.332924 

541 

642 

.333025 

.666975 

358   'XS  i  .999899 

46 

4500 

15 

.338753 

540 

644 

.338856 

.661144 

356   '°3 

.999897 

45 

4560 

16 

.344504 

539  1646 

.344610 

.655390 

354   -°5 

.999894 

44 

4620 

17 

.350181 

539  648 

.350289 

.649711 

352   '55 

.999891 

43 

4680 

18 

.355783 

538 

649 

.355895 

.644105 

351 

.uo 

.999888 

42 

4740 

19 

.361315 

537 

651 

.361430 

.638570 

349 

.05 

AK 

.999885 

41 

4800 

20 

.366777 

536 

653 

.366895 

.633105 

347 

.UO 

.999882 

40 

4860 

21 

8.372171 

535 

655 

8.372292 

11.627708 

345 

.05 

AC 

9.999879 

39 

4920 

22 

.377499 

534 

657 

.377622 

.622378 

343   -Jg 

.999876 

38 

4980 

23 

.382762 

533 

659 

.382889 

.617111 

341   •£ 

.999873 

37 

5040 

24 

.387962 

532 

661 

.388092 

.611908 

339   '« 

.999870 

36 

5100 

25 

.393101 

531  j 

663 

.393234 

.606766 

oor>    .UO 
ao7    nr; 

.999867 

35 

5160 

26 

.398179 

530; 

666 

.398315 

.601685 

334 

.UU 

'  0^ 

.999864 

34 

5220 

27 

.403199 

529| 

668 

.403338 

.596662 

332 

I  .UO 
AK 

.999861 

33 

5280 

28 

.408161 

527 

670 

.408304 

.591696 

330 

.UO 
CV7 

.999858 

32 

5340 

29 

.413068 

526 

672 

.413213 

.586787 

328 

.U< 
AK 

.998854 

31 

5400 

30 

.417919 

525 

674 

.418068 

.581932 

326 

.UO 

.999851 

30 

5460 

31 

8.422717 

524 

676 

8.422869 

11.577131 

324 

.05 

(V7 

9.999848 

29 

5520 

32 

.427462 

523 

679 

.427618 

.572382 

321 

•VI 

AK 

.999844 

28 

5580 

33 

.432156 

522 

681 

.432315 

.567685 

319 

.UO 
A'K 

.999841 

27 

5640 

34 

.436800 

521 

683 

.436962 

.563038 

317 

.UO 

Cf7 

.999838 

26 

5700 

35 

.441394 

520 

685 

.441560 

.558440 

315 

.VI 

AK 

.999834 

25 

5760 

36 

.445941 

518 

688 

.446110 

.553890 

312 

.UO 
(V7 

.999831 

24 

5820 

37 

.450440 

517 

690 

.450613 

.549387 

310 

.Ui 

i  fif* 

.999827 

23 

5880 

38 

.454893 

516! 

693 

.455070 

.544930 

307 

I  .UO 

.999824 

22 

5940 

39 

.459301 

515 

695 

.459481 

.540519 

305 

.07 

(Y7 

.999820 

21 

6000 

40 

.463665 

514 

697 

.463849 

.536151 

303 

.Ui 

.999816  i  20 

6060 

41 

8.467985 

512 

700 

8.468172 

11.531828 

300 

.05 

H7 

9.999813 

19 

6120 

42 

.472263 

511 

702 

.472454 

.527546 

298 

«U< 

(Y7 

.999809  18 

6180 

43 

.476498 

510 

705 

.476693 

.523307 

295 

.u< 

.999805  |  17 

6240 

44 

.480693 

509 

707 

.480892 

.519108 

293 

.07 

ray 

.999801   16 

6300 

45 

.484848 

50Y 

710 

.485050 

.514950 

290 

.U< 

.989797  i  15 

6360 

46 

.488963 

506 

713 

.489170 

.510830 

287 

.05 
cv* 

.999794  i  14 

6420 

47 

.493040 

505 

715 

.493250 

.506750 

285 

.u< 

On 

.999790 

13 

6480 

48 

.497078 

503 

718 

.497293 

.502707 

282 

i 
CV7 

.999786 

12 

6540 

49 

.501080 

502 

720 

.501298 

.498702 

280 

.U< 

07 

.999782 

11 

6600 

50 

.505045 

501 

723 

.505267 

.494733 

277 

.U< 

.999778  10 

6660 

51 

8.508974 

499 

726 

8.509200 

11.490800 

274 

.07 

AQ 

9.999774 

9 

6720 

52 

.512867 

498 

729 

.513098 

.486902 

271 

.UO 

O7 

.999769 

8 

6780 

53 

.516726 

497 

731 

.516961 

.483039 

269 

.Ui 

1  Qy 

.999765 

7 

6840 

54 

.520551 

495 

734 

.520790 

.479210 

266 

!   (W 

.999761 

6 

6900 

55 

.524343 

494 

737 

524586 

.475414 

263 

'  «U< 
CV7 

.999757 

5 

6960 

56 

.528102 

492 

740 

.528349 

.471651 

260 

.U< 
flft 

.999753 

4 

7020 

57 

.531828 

491 

743 

.532080 

.467920 

257 

.Uo 

A.7 

.999748 

3 

7080 

58 

.535523 

490; 

745 

.535779 

.464221 

255 

.U< 

O7 

.999744 

2 

7140 

59 

.539186 

488  i 

748 

.539447 

.460553 

252 

.U< 

AQ 

.999740 

1 

7200 

60 

8.542819  |487i 

751 

8.543084 

11.456916 

249 

•Uo 

9.999735 

0 

4.685 

15.314 

H 

/ 

Cosine. 

q-l 

Cotang. 

Tang. 

q  +  l 

Dl* 

Sine. 

'J 

91° 


105 


COSINES,   TANGENTS,   AND    COTANGENTS. 


' 

Sine. 

D.  r. 

Cosine. 

D.  1'. 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

2 
3 

4 
5 
6 

7 

8  542319 
.546422 
.549995 
.553539 
.557054 
.560540 
.563999 
.567431 

60.05 
59.55 
59.07 
58.58 
58.10 
57.65 
57.20 

9.999735 
.999731 
.999726 
.999722 
.999717 
.999713 
.999708 
.999704 

.07 
.08 
.07 
.08 
.07 
.08 
.07 

Aft 

8.543084 
.546691 
.550268 
.553817 
.557336 
.560828 
.564291 
.567727 

60.12 
59.62 
59.15 
58.65 
58.20 
57.72 
57.27 

f-O  Oq 

11.456916 
.453309 
.449732 
.446183 
.442664 
.439172 
.435709 
.432273 

60 
59 
58 
57 
56 
55 
54 
53 

8 

.570836 

56.75 

.999699 

.Uo 
no 

.571137 

00  .  OO 

tn  qo 

.428863 

52 

9 
10 

.574214 
.577566 

56.30 

55.87 
55.43 

.999694 
.999689 

.Uo 

.08 
.07 

.574520 

.577877 

OO  .  oo 

55.95 
55.52 

.425480 
.422123 

51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 

8.580892 
.584193 
.587469 
.590721 
.593948 
.597152 
.600332 
.603489 
.606623 

55.02 
54.60 
54.20 
53.78 
53.40 
53.00 
52.62 
52.23 

9.999685 
.999680 
.999675 
.999670 
.999665 
.999660 
.999655 
.999650 
.999645 

.08 
.08 
.08 
.08 
.08 
.08 
.08 
.08 
no 

8.581208 
.584514 
.587795 
.591051 
.594283 
.597492 
.600677 
.603839 
.606978 

55.10 
54.68 
54.27 
53.87 
53.48 
53.08 
52.70 
52.32 

11.418792 
.415486 
.412205 
.408949 
.405717 
.402508 
.399323 
.396161 
.393022 

49 
48 
47 
46 
45 
44 
43 
42 
41 

20 

.609734 

51  .85 
51.48 

.999640 

.Uo 
.08 

.610094 

51  .93 
51.58 

.389906 

40 

21 
22 
23 
24 
25 

8.612823 
.615891 
.618937 
.621962 
.624965 

51.13 
50.77 
50.42 
50.05 

9.999635 
.999629 
.999624 
.999619 
.999614 

.10 
.08 

.08 
.08 
in 

8.613189 
.616262 
.619313 
.622343 
.625352 

51.22 
50.85 
50.50 
50.15 

11.386811 
.383738 
.380687 
.377657 
.374648 

39 
38 
37 
36 
35 

26 
27 

28 

.627948 
.630911 
.633854 

49.72 
49.38 
49.  05. 

.999608 
.999603 
.999597 

.  JU 

.08 
.10 

AQ 

.628340 
.631308 
.634256 

49.80 
49.47 
49.13 

4ft  RO 

.371660 
.368692 
.365744 

34 
33 

32 

29 
30 

.636776 
.639680 

48^40 
48.05 

.999592 
.999586 

.UO 

.10 

.08 

.637184 
.640093 

4o.oU 

48.48 
48.15 

.362816 
.359907 

31 
30 

31 

8.642563 

9.999581 

8.642982 

41  Qfl 

11.357018 

29 

32 

.645428 

47.75 
4r»  jn 

.999575 

AO 

.645853 

f  .oO 

.354147 

28 

33 
34 
35 
36 
37 
38 
39 
40 

.648274 
.651102 
.653911 
.656702 
.659475 
.662230 
.664968 
.667689 

t  .4o 
47.13 
46.82 
46.52 
46.22 
45.92 
45.63 
45.35 
45.07 

.999570 
.999564 
.999558 
.999553 
.999547 
.999541 
.999535 
.999529 

.Uo 

.10 
.10 
.08 
.10 
.10 
.10 
.10 
.08 

.648704 
.651537 
.654352 
.657149 
.659928 
.662689 
.665433 
.668160 

47!  22 
46.92 
46.62 
46.32 
46.02 
45.73 
45.45 
45.17 

.351296 
.348463 
.345648 
.342851 
.340072 
.337311 
.334567 
.331840 

27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 

8.670393 
.673080 
.675751 
.678405 
.681043 
.683665 
.686272 
.688863 
.6914*^8 

44.78 
44.52 
44.23 
43.97 
43.70 
43.45 
43.18 
42.92 

9.999524 
.999518 
.999512 
.999506 
.999500 
.999493 
.999487 
.999481 
.999475 

.10 
.10 
.10 
.10 
.12 
.10 
.10 
.10 

8.670870 
.673563 
.676239 
.678900 
.681544 
.684172 
.686784 
.689381 
.691963 

44.88 
44.60 
44.35 
44.07 
43.80 
43.53 
43.28 
43.03 

11.329130 
.326437 
.323761 
.321100 
.318456 
.315828 
.313216 
.310619 
.308037 

19 
18 
17 
16 
15 
14 
13 
12 
11 

50 

.693998 

42.67 
42.42 

.999469 

.10 
.10 

.694529 

42.77 
42.53 

.305471 

10 

51 

52 
53 
54 
55 
56 
57 
58 
59 
60 

8.696543 
.699073 
.701589 
.704090 
.706577 
.709049 
.711507 
.713952 
.716383 
8.718800 

42.17 
41.93 
41.68 
41.45 
41.20 
40.97 
40.75 
40.52 
40.28 

9.999463 
.999456 
.999450 
.999443 
.999437 
.999431 
.999424 
.999418 
.999411 
9.999404 

.12 
.10 
.12 
.10 
.10 
.12 
.10 
.12 
.12 

8.697081 
.699617 
.702139 
.704646 
.707140 
.709618 
.712083 
.714534 
.716972 
8.719396 

42.27 
42.03 
41.78 
41.57 
41.30 
41.08 
40.85 
40.63 
40.40 

11.302919 
.300383 
.297861 
.295354 
.292860 
.290382 
.287917 
.285466 
.283028 
11.280604 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

* 

Cos^e. 

D.  r. 

Sine. 

D.  r. 

Cotang.  D.  1". 

Tang. 

' 

92" 


106 


87* 


TABLE    X. — LOGARITHMIC    SINES, 


' 

Sine. 

D.  1'. 

Cosine. 

D.  1*. 

Tang. 

D.I'. 

Cotang. 

' 

0 

1 

2 

3 
4 

8.718800 
.721204 
.723595 
.725972 

.728337 

40.07 
39.85 
39.62 
39.42 

9.999404 
.999398 
.999391 
.999384 
.999378 

.10 
.12 
.12 
.10 

19 

8.719396 
.721806 
.724204 
.726588 
.728959 

40.17 
39.97 
39.73 
39.52 

11.280604 
.278194 
.275796 
.273412 
.271041 

60 
59 
58 
57 
56 

6 

.730688 

qo  'QQ 

.999371 

.731317 

^o  in 

.2686aS 

55 

6 
7 
8 
9 
10 

.733027 
.735354 
.737667 
.739969 
.742259 

38.78 
38.55 
38.37 
38.17 
37.95 

.999364 
.999357 
.999350 
.999343 
.999336 

.12 
.12 
.12 
.12 
.12 

.733663 
.735996 
.738317 
.740626 
.742922 

38.88 
38.68 
38.48 
38.27 
38.08 

.266J337 
.264004 
.261683 
.259374 
.257078 

54 
53 
52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

8.744536 
.746802 
.749055 
.751297 
.753528 
.755747 
.757955 
.760151 
.762337 
.764511 

37.77 
37.55 
37.37 
37.18 
36.98 
36.80 
36.60 
36.43 
36.23 
36.07 

9.999329 
.999322 
.999315 
.999308 
.999301 
999294 
.999287 
.999279 
.999272 
.999265 

.12 
.12 
.12 
.12 
.12 
.12 
.13 
.12 
.12 
.13 

8.745207 
.747479 
.749740 
.751989 
.754227 
.756453 
.758668 
.760872 
.76:3065 
.765246 

37.87 
37.68 
37.48 
37.30 
37.10 
36.92 
36.73 
36.55 
36.35 
36.18 

11.254793 
.252521 
.250260 
.248011 
.245773 
.24:3547 
.241332 
.239128 
.2369:35 
.234754 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 

8.766675 

.768828 

a5.88 

qpr  r*n 

9.999257 
.999250 

.12 
1  ^ 

8.767417 
.769578 

36.02 

OR  QO 

11.232583 
.230422 

39 

38 

23 
24 
25 
26 

27 
28 
29 
30 

.770970 
.773101 
.775223 
.777333 
.779434 
.781524 
.783605 
.785675 

35.52 
35.37 
35.17 
35.02 
34.83 
34.68 
34.50 
34.35 

.999242 
.999235 
.999227 
.999220 
.999212 
.999205 
.999197 
.999189 

.12 
.13 
.12 
.13 
.12 
.13 
.13 
.13 

.771727 

.77:3866 
.775995 
.778114 
.780222 
.782320 
.784408 
.786486 

a5.65 
a5.48 
35.32 
86.18 

34.97 
34.80 
34.63 
34.47 

.228273 
.226134 
.224005 
.221886 
.219778 
.217680 
.215592 
.213514 

37  ^ 
36; 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

8.787736 
.789787 
.791828 
.793859 
.795881 
.797894 
.799897 
.801892 
.803876 
.805852 

34.18 
34.02 
as.  85 
33.70 

as.  55 

33.38 
33.25 

as.  07 

32.93 
32.78 

9.999181 
.999174 
.999166 
.999158 
.999150 
.999142 
.999134 
.999126 
.999118 
.999110 

.12 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 
.13 

8.788554 
.790613 
.792662 
.794701 
.796731 
.798752 
.800763 
.802765 
.804758 
.806742 

34.32 
34.15 

as.  98 
as.  as 

as.  68 

as.  52 

33.37 
33.22 

as.  07 

32.92 

11.211446 
.209387 
.207aS8 
.205299 
.203269 
.201248 
.199237 
.197'235 
.195242 
.193258 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 

8.807819 

9.999102 

8.808717 

rm 

11.1912aS 

19 

42 
43 
44 
45 

46 
47 
48 
49 

.809777 
.811726 
.813667 
.815599 
.817522 
.819436 
.821343 
.823240 

32.48 
32.35 
32.20 
32.05 
31.90 
31.78 
31.62 

.999094 
.999086 
.999077 
.999069 
.999061 
.999053 
.999044 
.999036 

.13 
.15 
.13 
.13 
.13 
.15 
.13 

.810683 
.812641 
.814589 
.816529 
.818461 
.820384 
.822298 
.824205 

32163 
32.47 

32.  as 

32.20 
32.05 
31.90 
31.78 

.189317 
.187359 
.185411 
.iaS471 
.181539 
.179616 
.177702 
.175795 

18 
17 
16 
15 
14 
13 
12 
11 

50 

.825130 

31.35 

.999027 

.13 

.826103 

31.48 

.173897 

10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

8.827011 
.828884 
.830749 
.832607 
.834456 
.836297 
.838130 
.839956 
.841774 
8.843585 

31.22 

31.08 
30.97 
30.82 
30.68 
30.55 
30.43 
30.30 
30.18 

9.999019 
.999010 
.999002 
.998993 
.998984 
.998976 
.998967 
.998958 
.998950 
9.998941 

.15 
.13 
.15 
.15 
.13 
.15 
.15 
.13 
.15 

8.827992 
.829874 
.831748 
.833613 
.835471 
.837321 
.839163 
.840998 
.842825 
8.844644 

31.37 
31  23 

31.08 
30.97 
30.83 
30.70 
30.58 
30.45 
30.32 

11.172008 
.170126 
,168252 
.166387 
.164529 
.162679 
.160837 
.159002 
.157175 
11.155356 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

'   Cosine. 

D  1". 

Sine. 

D.r. 

Cotang. 

D.  1".   Tang. 

' 

93° 


107 


COSINES,    TANGENTS,    AND    COTANGENTS. 


175° 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 
3 
4 
5 
6 

8 
9 
10 

8.843585 
.845387 
.847183 
.848971 
.850751 
.852525 
.854291 
.856049 
.857801 
.859546 
.861283 

30.03 
29.93 

29.80 
29.67 
29.57 
29.43 
29.30 
29.20 
29.08 
28.95 
28.85 

9.998941 
.998932 
.998923 
.998914 
.998905 
.998896 
.998887 
.998878 
.998869 
.998860 
.998851 

.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 
.15 
.17 

8.844644 
.846455 
.848260 
.850057 
.851846 
.853628 
.855403 
.857171 
.858932 
.860686 
.862433 

30.18 
30.08 
29.95 
29.82 
29.70 
29.58 
29.47 
29.35 
29.23 
29.12 
29.00 

ll.:f)5356 
.  153545 
.151740 
.149943 
.148154 
.146372 
.144597 
.142829 
.141068 
.139314 
.137567 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 

12 
13 
14 
15 
16 

8.863014 
.864738 
.866455 
.868165 
.869868 
.871565 

28.73 
28.62 
28.50 
28.38 
28.28 

9.998841 
.998832 
.998823 
.998813 
.998804 
.998795 

.15 
.15 
.17 
.15 
.15 

8.864173 
.865906 
.867632 
.869351 
.871064 
.872770 

28.88 
28.77 
28.65 
28.55 
28.43 

11.135827 
.134094 
.  132368 
.130649 
.128936 
.127230 

49 
48 
47 
46 
45 
44 

17 
18 
19 
20 

.873255 
.874938 
.876615 
.878285 

28.05 
27.95 
27  .-83 
27.73 

.998785 
.998776 
.998766 
.998757 

.15 

.17 
.15 
.17 

.874469 
.876162 
.  877849 
.879529 

28.22 
28.12 
28.00 
27.88 

.125531 
.123838 
.122151 
.120471 

43 
42 
41 
40 

21 

8.879949 

9.998747 

8.881202 

11.118798 

39 

22 
23 
24 
25 
26 
27 
28 
29 
30 

.881607 
.883258 
.884903 
.886542 
.888174 
.889801 
.891421 
.893035 
.894643 

27.52 
27.42 
27.32 
27.20 
27.12 
27.00 
26.90 
26.80 
26.72 

.998738 
.998728 
.998718 
.998708 
.998699 
.998689 
.998679 
.998669 
.998659 

.17 
.17 
.17 
.15 
.17 
.17 
.17 
.17 
.17 

.882869 
.884530 
.886185 
.887833 
.889476 
.891112 
.892742 
.894366 
.895984 

27.68 
27.58 
27.47 
27.38 
27  27 
27  '.17 
27.07 
26.97 
26.87 

.117131 
.115470 
.113815 
.112167 
.110524 
.108888 
.107258 
.105634 
.104016 

38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

8.896246 
.897842 
.899432 
.901017 
.902596 
.904169 
.905736 
.907297 
.908853 
.910404 

26.60 
26.50 
26.42 
26.32 
26.22 
26.12 
26.02 
25.93 
25.85 
25.75 

9.998649 
.998639 
.998629 
.998619 
.998609 
.998599 
.998589 
.998578 
.998568 
.998558 

.17 
.17 
.17 
.17 
.17 
.17 
.18 
.17 
.17 
.17 

8.897596 
.899203 
.900803 
.902398 
.903987 
.905570 
.907147 
.908719 
.910285 
.911846 

26.78 
26.67 
26.58 
26.48 
26.38 
26.28 
26.20 
26.10 
26.02 
25.92 

11.102404 
.100797 
.099197 
.097602 
.096013 
.094430 
.092853 
.091281 
.089715 
.088154 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 

42 

8.911949 
.913488 

25.65 

9.998548 
.998537 

.18 

8.913401 
.914951 

25.83 

11.086599 
.085049 

19 
18 

43 
44 
45 
46 

47 
48 
49 
50 

.915022 
.916550 
.918073 
.919591 
.921103 
.922610 
.924112 
.925609 

25.47 
25.38 
25.30 
25.20 
25.12 
25.03 
24.95 
24.85 

.998527 
.998516 
.998506 
.998495 
.998485 
.998474 
.998464 
.998453 

.18 
.17 
.18 
.17 
.18 
.17 
.18 
.18 

.916495 
.918034 
.919568 
.921096 
.922619 
.924136 
.925649 
.927156 

25.63 
25.57 
25.47 
25.38 
25.28 
25.22 
25.12 
25.03 

.083505 
.081966 
.080432 
.078904 
.077381 
.075864 
.074351 
.072844 

17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

8.927100 
.928587 
.930068 
.931544 
.933015 
.934481 
.935942 
.937398 
.938850 
8.940296 

24.78 
24.68 
24.60 
24.52 
24.43 
24.35 
24  27 
24^20 
24.10 

9.998442 
.998431 
.998421 
.998410 
.998399 
.998388 
.998377 
.998366 
.998355 
9.998344 

.18 
.17 
.18 
.18 
.18 
.18 
.18 
.18 
.18 

8.928658 
.930155 
.931647 
.933134 
.934616 
.936093 
.937565 
.939032 
.940494 
8.941952 

24.95 

24.87 
24.78 
24.70 
24.62 
24.53 
24.45 
24.37 
24.30 

11.071342 
.069845 
.068353 
.066866 
.065384 
.063907 
.062435 
.060968 
.059506 
11.058048 

9 

8 
7 
6 
5 

3' 
2 
1 
0 

' 

Cosine. 

D.  1". 

i  Sine. 

D.  1".  ! 

Cotang.  i  D.  1". 

Tang. 

' 

108 


85° 


TABLE    X. — LOGARITHMIC    SIXES, 


174° 


' 

Sin*. 

D.  1*.   Cosine. 

D.  r. 

Tang. 

D.  r. 

Cotang. 

- 

0 

1 

2 

3 
4 
5 
6 

8.940296 
.941738 
.943174 
.944606 
.946034 
.947456 
.948874 

24.03 
23.93 
23.87 
23.80 
23.70 
23.63 

OO  KK 

9.998344 
.998333 
.998322 
.998311 
.998300 
.998289 
.998277 

.18 
.18 
.18 
.18 
.18 
.20 
18 

8.941952 
.943404 
.944852 
.946295 
.947734 
.949168 
.950597 

24.20 
24.13 
24.05 
23.98 
23.90 
23.82 
23  73 

11.058048 
.056596 
.055148 
.053705 
.052266 
.050832 
.049403 

60 
59 
58 
57 
56 
55 
54 

7 
8 
9 
10 

.950287 
.951696 
.953100 
.954499 

0O.OO 

23.48 
23.40 
23.32 
23.25 

.998266 
.998255 
.998243 
.998232 

'.20 
.18 
.20 

.952021 
.953441 
.954856 
.956267 

23^67 
23.58 
23.52 
23.45 

.047979 
.046559 
.045144 
.043733 

53 
52 
51 
50 

11 

13 
14 
15 
16 
17 
18 
19 
20 

8.955894 
.957284 
.958670 
.960052 
.961429 
.962801 
.964170 
.965534 
.966893 
.968249 

23.17 
23.10 
23.03 
22.95 

22.87 
22.82 
22.73 
22.65 
22.60 
22.52 

9.998220 
.998209 
.998197 
.998186 
.998174 
.998163 
.998151 
.998139 
.998128 
.998116 

.18 
.20 
.18 
.20 
.18 
.20 
.20 
.18 
.20 
.20 

8.957674 
.959075 
.960473 
.961866 
.963255 
.964639 
.966019 
.967394 
.968766 
.970133 

23.35 
23.30 
23.22 
23.15 
23.07 
23.00 
22.92 
22.87 
22.78 
22.72 

11.042326 
.040925 
.039527 
.038134 
.036745 
.035361 
.033981 
.032606 
.031234 
.029867 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 

8.969600 
.970947 

22.45 

9.998104 
.998092 

.20 

90 

8.971496 
.972855 

22.65 

99  *f 

11.028504 
.027145 

39 
38 

23 
24 
25 

26 
27 
28 
29 
30 

.972289 
.973628 
.974962 
.976293 
.977619 
.978941 
.980259 
.981573 

22.37 
22.32 
22.23 
22.18 
22.10 
22.03 
21.97 
21.90 
21.83 

.998080 
.998068 
.998056 
.998044 
.998032 
.998020 
.998008 
.997996 

,4\) 

.20 
.20 
.20 
.20 
.20 
.20 
.20 
.20 

.974209 
.975560 
.976906 
.978248 
.979586 
.980921 
.982251 
.983577 

rC0.O< 

22.52 
22.43 
22.37 
22.30 
22.25 
22.17 
22.10 
22.03 

.025791 
.024440 
.023094 
.021752 
.020414 
.019079 
.017749 
.016423 

37 
36 
35 
34 
33 
32 
31 
30 

31 

32 

8.982883 
.984189 

21.77 

9.997984 
.997972 

.20 

8.984899 
.986217 

21.97 
21  92 

11.015101 
.013783 

29 
28 

33 
34 
35 
36 
37 
38 
39 
40 

.985491 
.986789 
.988083 
.989374 
.990660 
.991943 
.993222 
.994497 

21.72 
21.63 
21.57 
21.52 
21.43 
21.38 
21.32 
21.25 
21.18 

.997959 
.997947 
.997935 
.997922 
.997910 
.997897 
.997885 
.997872 

'.20 
.20 
.22 
.20 
.22 
.20 
.22 
.20 

.987532 
.988842 
.990149 
.991451 
.992750 
.994045 
.995337 
.996624 

2l!  83 
21.78 
21.70 
21.65 
21.58 
21.53 
21.45 
21.40 

.012468 
.011158 
.009851 
.008549 
.007250 
.005955 
.004663 
.003376 

27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 

8.995768 
.997036 
.  998299 
8.999560 

21.13 
21.05 
21.02 

0.997860 
.997847 
.997835 
.997822 

.22 
.20 
.22 

8.997908 
8.999188 
9.000465 
.001738 

21.33 

21.28 
21.22 

91  1  ^ 

11.002092 
11.000812 
10.999535 
.998262 

19 
18 
17 
16 

45 

9.000816 

20.93 

.997809 

Oft 

.003007 

if  L  .  ID 

.996993 

15 

46 
47 
48 
49 
50 

.002069 
.003318 
.004563 
.005805 
.007044 

20  88 
20.82 
20.75 
20.70 
20.65 
20.57 

.997797 
.997784 
.997771 
.997758 
.997745 

!22 
.22 
.22 
.22 
.22 

.004272 
.005534 
.006792 
.008047 
.009298 

21  .08 
21.03 
20.97 
20.92 
20.85 
20.80 

.995728 
.994466 
.993208 
.991953 
.990702 

14 
13 
12 
11 
10 

51 
52 
53 
54 
55 

9.008278 
.009510 
.010737 
.011962 
.013182 

20.53 
20.45 
20.42 
20.33 

9.997732 
.997719 
.997706 
.997693 
.997680 

.22 
.22 
.22 
.22 

9.010546 
.011790 
.013031 
.014268 
.015502 

20  73 
20.68 
20.62 
20  57 

10.989454 
.988210 
.986969 
.985732 
.984498 

9 

8 
7 
6 
5 

56 
!  57 

58 
59 
60 

.014400 
.015613 
.016824 
.018031 
9.019235 

20.30 
20.22 
20.18 
20.12 
20.07 

.P97667 
.997654 
!  .997641 
.  997628 
9.997614 

.22 

.22 
.22 
.22 
.23 

.016732 
.017959 
.019183 
.020403 
9.021620 

20^45 
20.40 
20.33 
20.28 

.983268 
.982041 
.980817 
.979597 
10.978380 

4 
3 
2 
1 
0 

'> 

Cosine. 

D.  r. 

!  Sine. 

D.  r. 

Cotang. 

D.  r. 

Tang. 

' 

95° 


109 


COSINES,  TANGENTS,  AND  COTANGENTS. 


/ 

Sine. 

D.r. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

• 

0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

9.019235 
.020435 
.021632 
.022825 
.024016 
.025203 
.026386 
.027567 
.028744 
.029918 
.031089 

20.00 
19.95 
19.88 
19.85 
19.78 
19.72  i 
19.68 
19.62 
19.57  j 
19  52 
19.47 

9.997614 
.997601 
.997588 
.997574 
.997561 
.997547 
.997534 
.997520 
.997507 
.997493 
.997480 

.22 
.22 
.23 
.22 
.23 
.22 
.23 
.22 
23 
'.22 
.23 

9.021620 
.0228134 
.024044 
.025251 
.026455 
.027655 
.028852 
.030046 
.031237 
.032425 
.033609 

20.23 
20.17 
20.12 
20.07 
20.00 
19.95 
19.90 
19.85 
19.80 
19.73 
19.70 

10.97&380 
.977166 
.975956 
.974749 
.973545 
.972345 
.971148 
.969954 
.968763 
.967575 
.966391 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 

9.032257 

1Q  4ft 

9.997466 

OO 

9.034791 

1Q  fi5l 

10.965209 

49 

12 
13 

14 
15 
16 
17 

18 
19 
20 

.033421 
.034582 
.035741 
.036896 
.038048 
.039197 
.040342 
.041485 
.042625 

iy  .4u 
19.35 
19.32 
19.25 
19.20 
19.15 
19.08 
19.05 
19.00 
18.95 

.997452 
.997439 
.997425 
.997411 
.997397 
.997383 
.997369 
.997355 
.997341 

,9H 

.22 
.23 
.23 
.23 
.23 
.23 
.23 
.23 
.23 

.035969 
.037144 
.038316 
.039485 
.040651 
.041813 
.042973 
.044130 
.045284 

jy  .00 
19.58 
19.53 
19.48 
19.43 
19.37 
19.33 
19.28 
19.23 
19.17 

.964031 
.962856 
.961684 
.960515 
.959349 
.958187 
.957027 
.955870 
.954716 

48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 

9.043762 
.044895 
.046026 

18.88 
18.85 

1ft  ftft 

9.997327 
.997313 
.997299 

.23 
.23 
23 

9.046434 
.047582 

.048727 

19.13 

19.08 

1Q  (Y^ 

10.953566 
.952418 
.951273 

39 

38 
37 

24 

.047154 

1O.OU 
1ft  ^ 

.997285 

.049869 

jy  .uo 

1ft  Qft 

.950131 

36 

25 

.048279 

Jo.  <O 

1ft  fift. 

.997271 

.23 

OQ 

.051008 

Jo.  yo 

1ft  Q^ 

.948992 

35 

26 

27 

.049400 
.050519 

Jo.Oo 

18.65 
1ft  fiO 

.997257 
.997242 

,X3 

.25 

oq 

.052144 
.053277 

j  o  .  yo 

18.88 
ift  ft1} 

.947856 
.946723 

34 
33 

28 

.051635 

1O.OI/ 

1ft  ^7 

.997228 

,110 

.054407 

Jo  oo 

1ft  ftfl 

.945593 

32 

29 
30 

.052749 
.053859 

Jo  .O< 

18.50 
18.45 

.997214 
.997199 

.23 

.25 
.23 

.055535 
.056659 

Jo.OU 

18.73 
18.70 

.944465 
.943341 

31 
30 

31 

32 
33 
34 
35 
36 
37 
38 
39 
40 

9.054966 
.056071 
.057172 
.058271 
.059367 
.060460 
.061551 
.062639 
.063724 
.064806 

18.42 
18.35 
18.32 
18.27 
18.22 
18.18 
18.13 
18.08 
18.03 
17.98 

9.997185 
.997170 
.997156 
.997141 
.997127 
.997112 
.997098 
.997083 
.997068 
.997053 

.25 
.23 
.25 
.23 
.25 
23 
!25 
.25 
.25 
.23 

9.057781 
.058900 
.060016 
.061130 
.062240 
.063348 
.064453 
.065556 
.066655 
.067752 

18.65 
18.60 
18.57 
18.50 
18.47 
18.42 
18.38 
18.32 
18.28 
18.25 

10.942219 
.941100 
.939984 
.938870 
.937760 
.936652 
.935547 
.934444 
.933345 
.932248 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 

45 
46 

47 
48 

9.065885 
.066962 
.068036 
.069107 
.070176 
.071242 
.072306 
.073366 

17.95 
17.90 
17.85 
17.82 
17.77 
17.73 
17.67 
17  63 

9.997039 
.997024 
.997009 
.996994 
.996979 
.996964 
.996949 
.996934 

.25 
.25 
.25 
25 

'.25 
.25 

OK 

9,068846 
.069938 
.071027 
.072113 
.073197 
.074278 
.075356 
.076432 

18.20 
18.15 
18.10 
18.07 
18.02 
17.97 
17.93 

1I~  QO 

10.931154 
.930062 
.928973 
.927887 
.926803 
.925722 
.924644 
.923568 

19 
18 
17 
16 
15 
14 
13 
12 

49 

.074424 

17  fift 

.996919 

OK 

.077505 

<  .OO 

1r»  QK 

.922495 

11 

50 

.075480 

1  1  .  OtF 

17.55 

.996904 

.^O 

.25 

.078576 

t  .OO 

17.80 

.921424 

10 

51 
52 
53 
54 

9.076533 
.077583 
.078631 
.079676 

17.50 
17.47 
17.42 

1r»  QQ 

9.996889 
.996874 
.996858 
.996843 

.25 
.27 
.25 

9.079644 
.080710 
.081773 
.082833 

17.77 
17.72 
17.67 

10.920356 
.919290 
.918227 
.917167 

9 

8 
7 
6 

55 

.080719 

<  .OO 

•  17  '-*'* 

.996828 

.27 

.083891 

17.63 

.916109 

5 

56 
57 
58 

.081759 
.082797 
.083832 

If  .OO 

17.30 
17.25 
17  on 

.996812 
.996797 
.996782 

.27 
.25 
25 

'9ff 

.084947 
.086000 
.087050 

17.60 
17.55 
17.50 

.915053 
.914000 
.912950 

4 
3 
2 

59 

60 

.084864 
9.085894 

i  ./vU 

17.17 

.996766 
5  9.996751 

'.25 

.088098 
9.089144 

17.47 
17.43 

.911902 
10.910856 

1 
0 

' 

Cosine. 

D.  1". 

Sine. 

D.  1". 

Cotang.  |  D.  1". 

Tang. 

' 

98° 


110 


TABLE    X. — LOGARITHMIC    SINES, 


172° 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang.  1  D.  r. 

Cotang. 

/ 

i 

0 

9  085894 

„ 

9.996751 

or- 

9.089144 

10.910856 

60 

1 

2 

3 
4 
5 

.086922 
.087947 
.088970 
.089990 
.091008 

17^08 
17.05 
17.00 
16.97 

.99(5735 
.996720 
.996704 
.996688 
.996673 

.25 
.27 
.27 
.25 

.090187 
.091228 
.092266 
.093302 
.094336 

17.38 
17.35 
17.30 
17.27 
17.23 

.909813 
.908772 
.907734 
.906098 
.905664 

59 
58 
57 
56 
55 

6 
7 
8 
9 
10 

.092024 
.093037 
.094047 
.095056 
.096062 

16.93 
16.88  ! 
16.83 
16.82 
16.77 
16.72 

.996657 
.996641 
.996625 
.996610 
.996594 

.27 
.27 
.27 
.25 
.27 
.27 

.095367 
.096395 
.097422 
.098446 
.099468 

17.18 
17.13 
17.12 
17.07 
17.03 
16.98 

.904633 
.903605 
.902578 
.901554 
.900532 

54 
53 
52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.097065 
.098066 
.099065 
.100062 
.101056 
.102048 
.103037 
.104025 
.105010 
.105992 

16.68 
16.65 
16.62 
16.57 
16.53 
16.48 
16.47 
16.42 
16.37 
16.35 

9.996578 
.996562 
.996546 
.996530 
.996514 
996498 
.996482 
.996465 
.996449 
.996433 

.27 
.27 
.27 
.27 
.27 
.27 
.28 
.27 
.27 
.27 

9.100487 
.101504 
.102519 
.103532 
.104542 
.105550 
.106556 
.107559 
.108560 
.109559 

16.95 
16.92 
16.88 
16.83 
16.80 
16.77 
16.72 
16.68 
16.65 
16.62 

10.899513 
.898496 
.897481 
.896468 
.895458 
.894450 
.893444 
.892441 
.891440 
.890441 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.106973 
.107951 
.108927 
.109901 
.110873 
.111842 
.112809 
.113774 
.114737 
.115698 

16.30 
16.27 
16.23 
16.20 
16.15  i 
16.12 
16.08 
16.05 
16.02 
15.97 

9.996417 
.996400 
.996384 
.996368 
.996351 
.996335 
.996318 
.996302 
.996285 
.996269 

.28 
.27 
.27 

.28 
.27 
.28 
.27 
.28 
.27 
.28 

9.110556 
.111551 
.112543 
.113533 
.  114521 
.115507 
.116491 
.117472 
.118452 
.119429 

16  58 
16.53 
16.50 
16.47 
16.43 
16.40 
16.35 
16.33 
16.28 
16.25 

10.889444 
.888449 
.887457 
.886467 
.885479 
.884493 
.883509 
.882528 
.881548 
.880571 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 

9  116656 

1  ^  Q*i  ' 

9.996252 

OQ 

9.120404 

10.879596 

29 

32 
33 
34 

.117613 
118567 
.119519 

10.  yo 
15.90 
15.87 

.996235 
.996219 
.996202 

.2o 

.27 

.28 

.121377 

.122348 
.123317 

16.22 
16.18 
16.15 

.878623 
.877652 
.876683 

28 
27 

35 
36 

.120469 
.121417 

15.83 
15.80 

.996185 
.996168 

.28 

.28 

.124284 
.125249 

16.12 
16.08 

.875716 

.874751 

25 
24 

37 
38 
39 

.122362 
.123306 
.124248 

15.75 
15.73 
15.70 

.996151 
.996134 
.996117 

.28 

,28 
.28 

.126211 
.127172 
.128130 

16.03 
16  02 
15.97 

.873789 
.872828 
.871870 

23 
22 
21 

40 

.  125187 

15.65 
15.63 

.996100 

.28 
.28 

.129087 

15.95 
15.90 

.870913 

20 

41 
42 
43 
44 

9  126125 
.127060 
.127993 
.128925 

15  58 
15.55 
15  53 

9.996083 
.996066 
.996049 
.996032 

.28 
.28 
.28 

9.130041 
.130994 
.131944 
.132893 

15.88 
15.83 

15.82 

10.869959 
.869006 
.868056 
.867107 

19 
18 
17 
16 

45 
46 
4? 
48 
49 
50 

.129854 
130781 
.131706 
.132630 
.133551 
.134470 

15.48 
15  45 
15.42 
15.40 
15  35 
15.32 
15.28 

.996015 
.995998 
.995980 
.995963 
.995946 
.995928 

.28 
.28 
.30 
.28 
.28 
.30 
.28 

.133839 
.134784 
.135726 
.136667 
.137605 
.138542 

15.77 
15.75 
15.70 
15.68 
15.63 
15  62 
15.57 

.866161 
.865216 
.864274 
.863333 
.862395 
.861458 

15 
14 
13 
12 

51 
52 
53 
54 
55 
56 

9  135387 
.136303 
.137216 
.  138128 
.139037 
.139944 

15  27 
15  22 
15.20 
15  15 
15.12 

^  995911 
.995894 
.995876 
.995859 
.995841 
.995823 

.28 
.30 
.28 
30 
.30 

9  139476 
.  140409 
.141340 
.142269 
143196 
.144121 

15.55 
15.52 
15.48 
15.45 
15.42 

10  860524 
.859591 
.858660 
.857731 
.856804 
.855879 

9 
8 
7 
6 
5 
4 

57 

58 
59 
60 

140850 
.141754 
.142655 
3  143555 

15.10 
15  07 
15.02 
15.00 

.995806 
.995788 
.995771 
9.995753 

.28 
.30 
28 
.30 

.145044 
.145966 
.  146885 
9.147803 

15.38 
15  37 
15.32 
15.30 

.854956 
.854034 
.853115 
10.852197 

3 

2 
1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  r. 

Cotang. 

D.  r. 

Tang. 

' 

111 


COSINES,   TANGENTS,  AND  COTANGENTS. 


171° 


/ 

Sine. 

D.  1". 

Cosine. 

D.  i'. 

Tang. 

D.  1". 

Cotang. 

/ 

0 

1 

2 
3 

9.143555 
.144453 
.145349 
.146243 

14.97 
14.93 
14.90 

HOQ 

9.995753 
.995735 
.995717 
.995699 

.30 
.30 
.30 
30 

9.147803 
.148718 
.  149632 
.150544 

15.25 
15.23 
15.20 

•f  c  17 

10.852197 
.851282 
.850368 
.849456 

60 
59 
58 
57 

4 

.147136 

.00 

.995681 

98 

.151454 

ID.  1  1 

-IK  -(  C 

.848546 

56 

5 
6 

7 
8 
9 

.148026 
.148915 
.  149802 
.150686 
.151569 

14.  83 
14.82 
14.78 
14.73 
14.72 

.995664 
.995646 
.995628 
.995610 
.995591 

.  4O 

.30 
.30 
.30 
.32 

OA 

.152363 
.153269 
.154174 
.155077 
.155978 

ID.  ID 

15.10 
15.08 
15.05 
15.02 

14  Q8 

.847637 
.846731 
.845826 
.844923 
.844022 

55 
54 
53 
52 
51 

10 

.152451 

14.70 
14.65 

.995573 

,oU 

.30 

.156877 

14.  yo 
14.97 

.843123 

50 

11 
12 
13 
14 

9.153330 
.154208 
.155083 
.155957 

14.63 
14.58 
14.57 

9.995555 
.995537 
.995519 
.995501 

.30 
.30 
.30 

9.157775 
.158671 
.159565 
.160457 

14.93 
14.90 
14.87 

14  8°. 

10.842225 
.841329 
.840435 
.839543 

49 
48 
47 
46 

15 
16 
17 
18 
19 
20 

.156830 
.157700 
.158569 
.159435 
.160301 
.161164 

14.55 
14.50 
14.48 
14.43 
14.43 
14.38 
14.35 

.995482 
.995464 
.995446 
.995427 
.995409 
.995390 

.32 
.30 
.30 
.32 

.30 
.32 
.30 

.161347 
.162236 
.163123 
.164008 
.164892 
.165774 

14.  OO 

14.82 
14.78 
14.75 
14.73 
14.70 
14.67 

.838653 
.837764 
.836877 
.835992 
.835108 
.834226 

45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 

9.162025 
.162885 
.163743 
.164600 
.165454 
.166307 
.167159 
.168008 

14.33 
14.30 
14.28 
14.23 
14.22 
14.20 
14.15 

9.995372 
.995353 
.995334 
.995316 
.995297 
.995278 
.995260 
.995241 

.32 
.32 
.30 
.32 
.32 
.30 
.32 

oo 

9.166654 
.167532 
.168409 
.169284 
.170157 
.171029 
.171899 
.172767 

14.63 
14.62 
14.58 
14.55 
14.53 
14.50 
14.47 

14  4^ 

10.833346 
.832468 
.831591 
.830716 

.829843 
.828971 
.828101 
.827233 

39 
38 
37 
36 
35 
34 
33 
32 

29 
30 

.168856 
.  169702 

14.13 
14.10 
14.08 

.995222 
.995203 

.0-6 

.32 
.32 

.173634 
.174499 

14  .  4D 

14.42 
14.38 

.826366 
.825501 

31 
30 

31 
32 
33 
34 

9.170547 
.171389 
.172230 
.173070 

14.03 
14.02 
14.00 

9.995184 
.995165 
.  995146 
.995127 

.32 
.32 
.32 

oo 

9.175362 
.176224 
.177084 
.177942 

14.37 
14.33 
14.30 

14  98 

10.824638 
.823776 
.822916 
.822058 

29 
28 
27 
26 

35 
36 
37 

.173908 
.174744 
.175578 

13.97 
13.93 
13.90 

•JO  QQ 

.995108 
.995089 
.995070 

,o4 

.32 
.32 
A9 

.178799 
.179655 
.180508 

14..CO 

14  27 
14.22 
14  20 

.821201 
.820345 
.819492 

25 
24 

23 

38 
39 
40 

.176411 
.177242 

.178072 

lO.  OO 

13.85 
13.83 
13.80 

.995051 
.995032 
.995013 

.oA 

.32 
.32 
.33 

.181360 
.182211 
.183059 

14.'  18 
14.13 
14.13 

.818640 
.817789 
.816941 

22 
21 
20 

41 
42 

9.178900 
.  179726 

13.77 

9.994993 
.994974 

.32 

qo 

9.183907 
.184752 

14.08 

14  rt8 

10.816093 
.815248 

19 
18 

43 
44 
45 
46 
47 

.180551 
.181374 
.182196 
.183016 
.183834 

13.75 
13.72 
13.70 
13.67 
13.63 

.994955 
.994935 
.994916 
.994896 
.994877 

.0* 

.33 
.32 
.33 
.32 

qq 

.185597 
.186439 
.187280 
.188120 
.188958 

14.  Uo 

14.03 
14.02 
14.00 
13.97 

.814403 
.813561 
.812720 
.811880 
.811042 

17 
16 
15 
14 
13 

48 
49 

.  184651 
.185466 

13.62 
13.58 

.994857 
.994838 

.00 

.32 

qq 

.189794 
.190629 

13.93 
13.92 

-«q  00 

.810206 
.809371 

12 
11 

50 

.186280 

13.57 
13.53  | 

.994818 

,OO 

.33 

.191462 

lo.oo 
13.87 

.808538 

10 

51 

9.187092 

1Q  &S>  ' 

9.994798 

qo 

9.192294 

13  83 

10.807706 

9 

52 
53 
54 
55 
56 
57 
58 
59 
60 

.187903 
.188712 
.189519 
.190325 
.191130 
.191933 
.192734 
.193534 
9.194332 

1  0  .  D(& 

13.48 
13.45 
13.43 
13.42 
13.38 
13.35 
13  33 
13  30 

.994779 
.994759 
.994739 
.994720 
.994700 
.994680 
.  994660 
.994640 
9.994620 

.o<« 
.33 
.33 
.32 
.33 
.33 
.33 
.33 
.33 

.193124 
.193953 
.194780 
.195606 
.  196430 
.197253 
.198074 
.198894 
9.199713 

13.  '82 
13.78 
13.77 
13.73 
13.72 
13.68 
13.67 
13.65 

.806876 
.806047 
.805220 
.804394 
.803570 
.802747 
.801926 
.801106 
10.800287 

8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine.   D.  1".  II  Sine. 

D.  1". 

Cotang. 

D.  1'. 

Tang. 

/ 

112 


81' 


TABLE    X. — LOGARITHMIC    SINES, 


170° 


' 

Sine. 

D.I". 

Cosine. 

D.  r. 

Tang. 

D.  r. 

Cotang. 

' 

0     9.194332 

13   Oft 

9.994620 

9.199713 

10.800287 

60 

1 
2 
3 
4 
5 
6 

.195129 
.195925 
.196719 
.197511 
.198302 
.199091 

13.27 
13.23 
13.20 
13.18 
13.15 

•jq  -jq 

.994600 
.994580 
.994560 
.994540 
.994519 
.994499 

.33    | 
.33    1 
.33    i 
.35 
.33    ! 

qq 

.200529 
.201345 
.202159 
.202971 
.203782 
.204592 

13.60 
13.57 
13.53 
13.52 
13.50 

•jq   Af-f 

.799471 
.798655 
.797841 
.797029 
.796218 
.795408 

59 
58 
57 
56 
55 
54 

7 

.199879 

-i  •»    i  .) 

.994479 

qq 

.205400 

•jq     Af 

.794600 

58 

8 
9 

.200666 
.201451 

13.08 

•jq   f\K 

.994459 
.994438 

.35 

qq 

.206207 
.207013 

13.43 

•jq     Af) 

.793793 

.792987 

52 
51 

10 

.202234 

13.05 

.994418 

.33 

.207817 

13.37 

.792183 

50 

11 

18 
14 
15 
16 

9.203017 
.203797 
.204577 
.205354 
.206131 
.206906 

13.00 
13.00 
12.95 
12.95 
12.92 

9.994398 
.994377 
.994357 
.994336 
.994316 
.994295 

.35     ! 
.33 
.35 
.33    ] 
.35 

qt 

9.208619 
.209420 
.210220 
.211018 
.211815 
.212611 

13.35 
13.33 
13.30 
13.28 
13.27 

1Q    OQ 

10.791381 
.790580 
.789780 
.788982 
.788185 
.787389 

49 
48 
47 
46 
45 
44 

17 
18 
19 
20 

.207679 
.208452 
.209222 
.209992 

12.88 
12.83 
12.83 
12.80 

.994274 
.994254 
.994233 
.994212 

.33 
.35    ! 
.35 
.35 

.213405 
.214198 
.214989 
.215780 

13.22 
13.18 
13.18 
13.13 

.786595 
.785802 
.785011 
.784220 

43 
42 
41 
40 

21 

9.210760 

19  77 

9.994191 

qq 

9.216568 

•jq  -jo 

10.783432 

39 

22 
23 
24 
25 
26 
27 
28 
29 
30 

.211526 
.212291 
.213055 
.213818 
.214579 
.215338 
.216097 
.216854 
.217609 

12.75 
12.73 
12.72 
12.68 
12.65 
12.65 
12.62 
12.58 
12.57 

.994171 
.994150 
.994129 
.994108 
.994087 
.994066 
.994045 
.994024 
.994003 

.35 
.35 
.35 
.35 
.35 
.35 
.35 
.35 
.35 

.217356 
.218142 
.218926 
.219710 
.220492 
.221272 
.222052 
.222830 
.223607 

13.10 
13.07 
13.07 
13.03 
13.00 
13.00 
12.97 
12.95 
12.92 

.782644 
.78ia58 
.781074 
.780290 
.779508 
.778728 
.777948 
.777170 
.776393 

38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.218363 
.219116 
.219868 
.220618 
.221367 
.222115 
.222861 
.223606 
.224349 
.225092 

12.55 
12.53 
12.50 
12.48 
12.47 
12.43 
12.42 
12.38 
12.38 
12.35 

9.993982 
.993960 
.993939 
.993918 
.993897 
.993875 
.993854 
.993832 
.993811 
.993789 

.37 
.35 
.35 
.35 
.37 
.35 
.37 
.35 
.37 
.35 

9.224382 
.225156 
.225929 
.226700 
.227471 
.228239 
.229007 
.229773 
.230539 
.231302 

12.90 
12.88 
12.85 
12.85 
12.80 
12.80 
12.77 
12.77 
12.72 
12.72 

10.775618 
.774844 
.  774071 
.773300 
.772529 
.771761 
.770993 
.770227 
.769461 
.768698 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 

44 
45 
46 
47 
48 
49 
50 

9.225833 
.226573 
.227311 

.228048 
.228784 
.229518 
.230252 
.230984 
.231715 
.232444 

12.33 
12.30 
12.28 
12.27 
12.23 
12.23 
12.20 
12.18 
12.15 
12.13 

9.993768 
.993746 
.993725 
.993703 
.993681 
.993660 
.993638 
.993616 
.993594 
.993572 

.37 
.35 
.37 
.37 
.35 
.37 
.37 
.37 
.37 
.37 

9.232065 
.232826 
.233586 
.234345 
.235103 
.235859 
.236614 
.237368 
.238120 
.238872 

12.68 
12.67 
12.65 
12.63 
12.60 
12.58 
12.57 
12.53 
12.53 
12.50 

10.767935 
.767174 
.766414 
.76565* 
.764897 
.764141 
.76a386 
.762632 
.761880 
.761128 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 

52 
53 
54 
55 
56 
57 
58 
59 
60 

9.233172 
.233899 
.234625 
.235349 
.236073 
.236795 
.237515 
.238235 
.238953 
9.239670 

12.12 
12.10 
12.07 
12.07 
12.03 
12.00 
12.00 
11.97 
11.95 

9.993550 
.993528 
.993506 
.993484 
.993462 
.993440 
.993418 
.993396 
.993374 
9.993351 

.37 
.37 
.37 
.37 
.37 
.37 
.37 
.37 
.38 

9.239622 
.240371 
.241118 
.241865 
.242610 
.243354 
.244097 
.244839 
.245579 
9.246319 

12.48 
12.45 
12.45 
12.42 
12.40 
12.38 
12.37 
12.33 
12.33 

10.760378 
.759629 
.758882 
.758135 
.757390 
.756646 
.755903 
.755161 
.754421 
10.753681 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine.      D.  1". 

Sine. 

D.  r. 

Cotang. 

D.  1". 

Tang. 

1 

09° 


113 


COSINES,  TANGENTS,  AND  COTANGENTS. 


169" 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  r. 

Cotang. 

' 

0 

9.239670 

9.993351 

q7 

9.246319 

10.753681 

60 

1 

.240386 

fr*2S 

.99:3329 

.Of 

.247057 

12.30 

.752943 

59 

2 

.241101 

11  .\r-l 

.993307 

.37 

.247794 

12.28 

.752206 

58 

3 

.241814 

11  .88 

.993284 

.38 

.248530 

12.27 

.751470 

57 

4 

.242526 

11.87 

nOpr 

.993262 

.37 

.249264 

12.23 

.750736 

56 

5 
6 

8 
9 
10 

.243237 
.243947 
.244656 
.245363 
.246069 
.246775 

.OO 

11.83 
11.82 
11.78 
11.77 
11.77 
11.72 

.993240 
.993217 
.993195 
.993172 
.993149 
.993127 

.37 
.38 
.37 
.38 
.38 
.37 
.38 

.249998 
.250730 
.251461 
.252191 
.252920 
.253648 

12.23 
12.20 
12.18 
12.17 
12.15 
12.13 
12.10 

.750002 
.749270 
.748539 
.747809 
.747080 
.746352 

55 
54 
53 
52 
51 
50 

11 
12 
13 
14 
15 
16 
17 

9.247478 
.248181 
.248883 
.249583 
.250282 
.250980 
.251677 

11.72 
11.70 
11.67 
11.65 
11.63 
11.62 

nfift 

9.993104 
.993081 
.993059 
.993036 
.993013 
.992990 
.992967 

.38 
.37 
.38 
.38 
.38 
.38 

qo 

9.254374 
.255100 
.255824 
.256547 
.257269 
.257990 
.258710 

12.10 
12.07 
12.05 
12.03 
12.02 
12.00 

10.745626 
.744900 
.744176 
.743453 
.742731 
.742010 
.741290 

49 

48 
47 
46 
45 
44 
43 

18 
19 

.252373 
.253067 

.ou 
11.57 

.992944 
.992921 

.GO 

.38 

.259429 
.260146 

11.98 
11.95 

.740571 
.739854 

42 
41 

20 

.253761 

11.57 
11.53 

.992898 

.38 

.38 

.260863 

11.95 
11.92 

.739137 

40 

21 

9.254453 

nKO 

9.992875 

9.261578 

10.738422 

39 

22 

.255144 

.vffs 

.992852 

.38 

.262292 

11.90 

.737708 

38 

23 
24 
25 
26 

.255834 
.256523 
.257211 

.257898 

11  .50 
11.48 
11.47 
11.45 

.992829 
.992806 
.992783 
.992759 

.38 
.38 
.38 
.40 

.263005 
.263717 
.264428 
.265138 

11.88 
11.87 
11.85 
11.83 

.736995 
.736283 
.735572 

.734862 

37 
36 
35 
34 

27 

28 

.258583 
.259268 

11  .42 
11.42 

nqo 

.992736 
.992713 

.38 
.38 

.265847 
.266555 

11.82 
11.80 

.734153 
.733445 

33 
32 

29 
30 

.259951 
.260633 

.OO 

11.37 
11.35 

.992690 
.992666 

.38 
.40 
.38 

.267261 
.267967 

11.77 
11.77 
11.73 

.732739 
.732033 

31 
30 

31 

32 
33 
34 
35 
36 
37 
38 
39 
40 

9.261314 
.261994 
.262673 
.263351 
.264027 
.264703 
.265377 
.266051 
.266723 
.267395 

11.33 
11.32 
11.30 
11.27 
11.27 
11.23 
11.23 
11.20 
11.20 
11.17 

9.992643 
.992619 
.992596 
.992572 
.992549 
.992525 
.992501 
.992478 
.992454 
.992430 

.40 
.38 
.40 
.38 
.40 
.40 
.38 
.40 
.40 
.40 

9.268671 
.269375 
.270077 
.270779 
.271479 
.272178 
.272876 
.273573 
.274269 
.274964 

11.73 
11.70 
11.70 
11.67 
11.65 
11.63 
11.62 
11.60 
11.58 
11.57 

10.731329 
.730625 
.729923 
.729221 
.728521 
.727822 
.727124 
.726427 
.725731 
.725036 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 

42 

9.268065 
.268734 

11.15 

9.992406 
.992382 

.40 

9.275658 
.276351 

11.55 

10.724342 
.723649 

19 

18 

43 
44 

45 
46 

47 

.269402 
.270069 
.270735 
.271400 
.272064 

Il!l2 
11.10 
11.08 
11.07 

nrjq 

.992359 
.992335 
.992311 
.992287 
.992263 

.38 
.40 
.40 
.40 
.40 

.277043 
.277734 

.278424 
.279113 
.279801 

11.53 
11.52 
11.50 
11.48 
11.47 

.722957 
.722266 
.721576 

.720887 
.720199 

17 
16 
15 
14 
13 

48 

.272726 

.Uo 

.992239 

.40 

.280488 

11.45 

.719512 

12 

49 
50 

.273388 
.274049 

11.03 

11.02 
10.98 

.992214 
.992190 

.42 

.40 

.40 

.281174 
.281858 

11.43 
11.40 
11.40 

.718826 
.718142 

11 
10 

51 

9.274708 

9.992166 

9.282542 

10.717458 

9 

52 
53 
54 
55 
56 
57 
58 
59 
.60 

.275367 
.276025 
.276681 
.277337 
.277991 
.278645 
.279297 
.279948 
9.280599 

10  '.97 
10.93 
10.93 
10.90 
10.90 
10.87 
10.85 
10.85 

.992142 
.992118 
.992093 
.992069 
.992044 
.992020 
.991996 
.991971 
9.991947 

.40 
.40 
.42 
.40 
.42 
.40 
.40 
.42 
.40 

.283225 
.283907 
.284588 
.285268 
.285947 
.286624 
.287301 
.287977 
9.288652 

11.38 
11.37 
11.35 
11.33 
11.32 
11.28 
11.28 
11.27 
11.25 

.716775 
.716093 
.715412 
.714732 
.714053 
.713376 
.712699 
.712023 
10.711348 

8 
7 
6 
5 
4 
3 
2 
3 
0 

1 

Cosine. 

D.  1".  1 

Sine.   D.  1". 

Cotang. 

D.I'. 

Tang,  i  ' 

114 


79* 


11* 


TABLE    X. — LOGARITHMIC    SINES, 


168* 


' 

Sine. 

D.  1". 

Cosine. 

D.  1'. 

Tang. 

I).  1". 

Cotang. 

' 

0 

1 

2 

3 
4 
5 
6 
7 
8 
9 
10 

9.280599 
.281248 
.281897 
.282544 
.283190 
.283836 
.284480 
.285124 
.285766 
.286408 
.287048 

10.82 
10.82 
10.78 
10.77 
10.77 
10.73 
10.73 
10.70 
10.70 
10.67 
10.67 

9.991947 
.991922 
.991897 
.991873 
.991848 
.991823 
.991799 
.991774 
.991749 
.991724 
.991699 

42 
'.42 
.40 
.42 
.42 
.40 
.42 
.42 
.42 
.42 
.42 

9.288652 
.289326 
.289999 
.290671 
.291342 
.292013 
.292682 
.293350 
.294017 
.294684 
.295349 

11.23 
11.22 
11.20 
11.18 
11.18 
11.15 
11.13 
11.12 
11.12 
11.08 
11.07 

10.711348 
.710674 
.710001 
.709329 
.708658 
.7'07987 
.707318 
.706650 
.705963 
.705316 
.704651 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 

9.287688 

9.991674 

9.296013 

10.703987 

49 

12 

.288326 

JJJ*?5 

.991649 

•jj 

.296677 

.703323 

48 

13 

.288964 

.991624 

.297339 

.702661 

47 

14 
15 
16 
17 
18 
19 
20 

.289600 
.290236 
.290870 
.291504 
.292137 
.292768 
.293399 

10.60 
10.57 
10.57 
10.55 
10.52 
10.52 
10.50 

.991599 
.991574 
.991549 
.991524 
.991498 
.991473 
.991448 

.42 
.42 
.42 
.43 
.42 
.42 
.43 

.298001 
.298662 
.299322 
.299980 
.300638 
.301295 
.301951 

11.03 
11.02 
11.00 
10.97 
10.97 
10.95 
10.93 
10.93 

.701999 
.701338 
.700678 
.700020 
.699362 
.698705 
.698049 

46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.294029 
.294658 
.295286 
.295913 
.296539 
.297164 
.297788 
.298412 
.289034 
.299655 

10.48 
10.47 
10.45 
10.43 
10.42 
10.40 
10.40 
10.37 
10.35 
10.35 

9.991422 
.991397 
.991372 
.991346 
.991321 
.991295 
.991270 
.991244 
.991218 
.991193 

.42 
.42 
.43 
.42 
.43 
.42 
.43 
.43 
.42 
.43 

9.302607 
.303261 
.303914 
.304567 
.305218 
.305869 
.306519 
.307168 
.307816 
.308463 

10.90 
10.88 
10.88 
10.85 
10.85 
10.83 
10.82 
10.80 
10.78 
10.77 

10.697393 
.696739 
.696086 
.695433 
.694782 
.694131 
.693481 
.692832 
.692184 
.691537 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 

32 

9.300276 
.300895 

10.32 

9.991167 
.991141 

.43 

9.309109 
.309754 

10.75 

10.690891 
.690246 

29 

28 

33 
34 
35 

36 
37 

.301514 
.302132 
.302748 
.303364 
.303979 

10.30 
10.27 
10.27 
10.25 

.991115 
.991090 
.991064 
.991038 
.991012 

.43 
.42 
.43 
.43 
.43 

.310399 
.311042 
.311685 
.312327 
.312968 

10.75 
10.72 
10.72 
10.70 
10.68 

.689601 
.688958 
.688315 
.687673 
.687032 

27 
26 
25 
24 
23 

38 
39 
40 

.304593 
.305207 
.305819 

10.23 
10.23 
10.20 
10.18 

.990986 
.990960 
.990934 

.43 
.43 
.43 
.43 

.313608 
.314247 
.314885 

10.67 
10.65 
10.63 
10.63 

.686392 
.685753 
.685115 

22 
21 
20 

41 
42 
43 

44 
45 
46 
47 
48 
49 
50 

9.306430 
.307041 
.307650 
.308259 
.308867 
.309474 
.310080 
.310685 
.311289 
.311893 

10.18 
10.15 
10.15 
10.13 
10.12 
10.10 
10.08 
10.07 
10.07 
10.03 

i  9.990908 
1  .990882 
.990855 
!  .990829 
.990803 
.990777 
.990750 
.990724 
.990697 
.990671 

.43 
.45  ! 
.43 
.43 
.43 
.45 
.43 
.45 
.43 
.43 

9.315523 
.316159 
.316795 
.317430 
.318064 
.318697 
319330 
.319961 
.320592 
.321222 

10.60 
10.60 
10.58 
10.57 
10.55 
10.55 
10.52 
10.52 
10.50 
10.48 

10.684477 

'.  683205 
.682570 
.681936 
.681303 
.680670 
.680039 
.679408 
.678778 

19 

18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
65 
56 
57 
58 
59 
60 

9.312495 
.313097 
.313698 
.314297 
.314897 
.315495 
.316092 
.316689 
.317284 
9.317879 

10.03 
10.02 
9.98 
10.00 
9.97 
9.95 
9.95 
9.92 
9.92 

9.990645 
.990618 
.990591 
.990565 
.990538 
.990511 
.990485 
.990458 
.990431 
9.990404 

.45 
.45 
.43 
.45 
.45 
.43 
.45 
.45 
.45 

9.321851 
.322479 
.323106 
.323733 
.324358 
.324983 
.325607 
.326231 
.326853 
9.327475 

10.47 
10.45 
10.45 
10.42 
10.42 
10.40 
10.40 
10.37 
10.37 

10.678149 
.677521 
.676894 
.676267 
.675642 
.675017 
.674393 
.673769 
.673147 
10.672525 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  1". 

Cotang. 

D.  r. 

Tang.   ' 

115 


COSINES,  TANGENTS,  AND  COTANGENTS. 


' 

Sine. 

D.  1". 

Cosine. 

D.  1'. 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 
3 
4 
5 
6 
7 

9.317879 
.318473 
.319066 
.319658 
.320249 
.320840 
.321430 
.322019 

9.90 
9.88 
9.87 
9.85 
9.85 
9.83 

9.82 
9  on 

9.990404 
.990378 
.990351 
.990324 
.990297 
.990270 
.990243 
.990215 

.43 
.45 
.45 
.45 
.45 
.45 
.47 

9.327475 
.328095 
.328715 
.329334 
.329953 
.330570 
.331187 
.331803 

10.33 
10.33 
10.32 
10.32 
10.28 
10.28 
10.27 

10.672525 
.671905 
.671285 
.670666 
.670047 
.669430 
.668813 
.668197 

60 
59 
58 
57 
56 
55 
54 
53 

8 
9 
10 

.322607 
.323194 
.323780 

.8U 
9.78 
9.77 
9.77 

.990188 
.990161 
.990134 

.45 
.45 
.45 
.45 

.332418 
.333033 
.333646 

10.25 
10.25 
10.22 
10.22 

.667582 
.666967 
.666354 

52 
51 
50 

11 
12 
13 
14 
15 
16 

9.324366 
.324950 
.325534 
.326117 
.326700 
.327281 

9.73 
9.73 
9.72 
9.72 

9.68 

9  CO 

9.990107 
.990079 
.990052 
.990025 
.989997 
.989970 

.47 
.45 
.45 
.47 
.45 

47 

9.334259 
.834871 
.335482 
.336093 
.336702 
.337311 

10.20 
10.18 
10.18 
10.15 
10.15 
10  1  ^ 

10.665741 
.665129 
.664518 
.663907 
.663298 
.662689 

49 
48 
47 
46 
45 
44 

17 
18 
19 
20 

.327862 
.328442 
.329021 
.329599 

.Do 

9.67 
9.65 
9.63 
9.62 

.989942 
.989915 
.989887 
.989860 

.4< 

.45 
.47 
.45 

.47 

.337919 
.338527 
.339133 
.339739 

JU.  Id 

10.13 
10.10 
10.10 
10.08 

.662081 
.661473 
.660867 
.660261 

43 

42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 

9.330176 
.330753 
.831329 
.331903 
.332478 
.333051 
.333624 
.334195 
.334767 

9.62 
9.60 
9.57 
9.58 
9.55 
9.55 
9.52 
9.53 

9.989832 
.989804 
.989777 
.989749 
.989721 
.989693 
.989665 
.989637 
.989610 

.47 
.45 
.47 
.47 
.47 
.47 
.47 
.45 

9.340344 
.340948 
.341552 
.342155 
.342757 
.343358 
.343958 
.344558 
.345157 

10.07 
10.07 
10.05 
10.03 
10.02 
10.00 
10.00 
9.98 

10.659656 
.659052 
.658448 
.657845 
.657243 
.656642 
.656042 
.655442 
.654843 

39 
38 
37 
36 
35 
34 

as 

32 
31 

30 

.335337 

9.50 
9.48 

.989582 

.47 
.48 

.345755 

9.97 
9.97 

.654245 

30 

31 

9.335906 

940 

9.989553 

9.346353 

9  93 

10.653647 

29 

32 
33 

.336475 
.337043 

.'to 

9.47 

.989525 
.989497 

!47 

.346949 
.347545 

9^93 

9QO 

.653051 
.652455 

28 
27 

34 

.337610 

9.45 

.989469 

.47 

.348141 

.yo 

.651859 

26 

35 

.338176 

9.43 

.989441 

.47 

.348735 

9.90 

.651265 

25 

36 

.338742 

9.43 

.989413 

.47 

.349329 

9.90 

9dd 

.650671 

24 

37 

.339307 

9.42 

.989385 

.47 

.349922 

.08 
907 

.650078 

23 

38 
39 
40 

.339871 
.340434 
.340996 

9^38 
9.37 
9.37 

.989356 
.989328 
.989300 

!47 
.47 
.48 

.350514 
.351106 
.351697 

.8< 

9.87 
9.85 
9.83 

.649486 
.648894 
.648303 

22 
21 
20 

41 

9.341558 

9  OK 

9.989271 

9.352287 

900 

10.647713 

19 

42 
43 

.342119 
.342679 

.OO 

9.33 

9OQ 

.989243 
.989214 

.47 

.48 

.352876 
.353465 

.8/4 

9.82 

9  Of) 

.647124 
.646535 

18 
17 

44 
45 

.343239 
.343797 

.OO 

9.30 

.989186 
.989157 

.47 
.48 

Ad 

.354053 
.354640 

.8U 

9.78 

970 

.645947 
.645360 

16 
15 

46 

.344355 

9  OS 

.989128 

.48 

.355227 

.  (8 

.644773 

14 

47 
48 
49 
50 

.344912 
.345469 
.346024 
.346579 

.60 
9.28 
9.25 
9.25 
9.25 

.989100 
.989071 
.989042 
.989014 

.47 

.48 
.48 
.47 
.48 

.355813 
.356398 
.356982 
.357566 

9^75 
9.73 
9.73 
9.72 

.644187 
.643602 
.643018 
.642434 

13 
12 
11 
10 

51 
52 
53 
54 
55 
56 

9.347134 
.347687 
.348240 
.348792 
.349343 
.349893 

9.22 
9.22 
9.20 
9.18 
9.17 

917 

!  988956 
.988927 
.988898 
.988869 
..988840 

.48 
.48 
.48 
.48 
.48 

Ad 

9.358149 
.358731 
.359313 
.359893 
.360474 
.361053 

9.70 
9.70 
9.67 
9.68 
9.65 

9  OK 

10.641851 
.641269 
.640687 
.640107 
.639526 
.638947 

9 
8 
7 
6 
5 
4 

57 
58 
59 

.350443 
.350992 
.351540 

.  1  < 

9.15 
9.13 
91°. 

.988811 
.988782 
.988753 

.48 

.48 
.48 

Ad 

.361632 
.362210 
.362787 

.DO 

9.63 
9.62 

Q  fiO 

.638368 
.637790 
.637213 

3 
2 

1 

60 

9.352088 

.  Id 

9.988724 

.48 

9.3G3364 

'  10.636636 

0 

' 

Cosine.   D.  r. 

Sine, 

D.  r. 

1  Cotang. 

D.  1".   Tang. 

' 

102* 


116 


13° 


TABLE    X. — LOGARITHMIC    SINES, 


166- 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

9.352088 
.352635 

9.12 
9  10 

9.988724 
.988695 

.48 

9.363364 
.363940 

9.60 

10.636636 
.636060 

60 
59 

2 
3 

4 
5 
6 

7 

.353181 
.353726 
.354271 
.354815 
.355358 
.355901 

9^08 
9.08 
9.07 
9.05 
9.05 

.988666 
.988636 
.988607 
.988578 
.988548 
.988519 

!so 

.48 
.48 
.50 

.48 

.364515 
.365090 
.365664 
.366237 
.366810 
.367382 

9.58 
9.58 
9.57 
9.55 
9.55 
9.53 

.635485 
.634910 
.634336 
.633763 
.633190 
.632618 

58 
57 
56 
55 
54 
53 

8 
9 
10 

.356443 
.356984 
.357524 

9.02 
9.00 

.988489 
.988460 
.988430 

.50 
.48 
.50 

.367953 
.368524 
.369094 

9.52 
9.52 
9.50 

.632047 
.631476 
.630906 

52 
51 
50 

9.00 

.48 

9.48 

11 
12 

9.358064 
.358603 

8.98 

9.988401 
.988371 

.50 

9.369663 
.370232 

9.48 

10.630337 
.629768 

49 

48 

13 

.359141 

8.97 

8QK 

.988342 

.48 

.370799 

9.45 

.629201 

47 

14 

.359678 

.yo 

8  or; 

.988312 

.50 

.371367 

9.47 

9   Aft 

.628633 

46 

15 
16 
17 

18 
19 

.360215 
.360752 
.361287 
.361822 
.362356 

.yo 
8.95 
8.92 
8.92 
8.90 

O  QO 

.988282 
.988252 
.988223 
.988193 
.988163 

'.50 
.48 
.50 
.50 

.371933 
.372499 
.373064 
.373629 
.374193 

.40 
9.43 
9.42 
9.42 
9.40 

9OQ 

.628067 
.627501 
.626936 
.626371 
.625807 

45 
44 
43 
42 
41 

20 

.362889 

O.OO 

8.88 

.988133 

!50 

.374756 

.OO 

9.38 

.625244 

40 

21 
22 

9.363422 
.363954 

8.87 

8  OK 

9.988103 

.988073 

.50 

9.375319 
.375881 

9.37 

10.624681 
.624119 

39 

38 

23 

.364485 

.DO 

.988043 

.50 

trv 

.376442 

9.35 

Q  Q^ 

.623558 

37 

24 
25 
26 

.365016 
.365546 
.366075 

8^83 

8.82 
809 

.988013 
.9879813 
.987953 

.OU 

.50 
.50 

.377003 
.377563 
.378122 

y  .00 
9.33 
9.32 

.622997 
.622437 
.621878 

36 
35 
84 

27 

.366604 

.o/* 

8r»o 

.987922 

.52 

.378681 

O  QO 

.621319 

33 

28 
29 
30 

.367131 
.367659 
.368185 

.  t  O 

8.80 
8.77 

.987892 
.987862 
.987832 

!so 

.50 

.379239 
.379797 
.380354 

s>!ao 

9.28 

.620761 
.620203 
.619646 

32 
31 
30 

8.77 

.52 

9.27 

31 

9.368711 

8r*K 

9.987801 

9.380910 

q  o« 

10.619090 

29 

32 
33 
34 
35 

.369236 
.369761 
.370285 

.370808 

.  i  •> 

8.75 

8.72 
8.72 

.987771 
.987740 
.987710 
.987679 

!52 

.50 
.52 

.381466 
.382020 
.382575 
.383129 

9^23 
9.25 
9.23 

.618534 
.617980 
.617425 
.616871 

28 
27 
26 
25 

36 
37 

38 

.371330 
.371852 
.372373 

8!70 
8.68 

.987649 
.987618 

.987588 

!52 
.50 

.383682 
.384234 
.384786 

9^20 
9.20 

.616318 
.615766 
.615214 

24 
23 
22 

39 
40 

.372894 
.373414 

8.68 
8.67 
8.65 

.987557 
.987526 

!52 
.50 

.385337 

.385888 

9.18 
9.18 
9.17 

.614663 
.614112 

21 

20 

41 
42 
43 
44 
45 

9.373933 
.374452 
.374970 
.375487 
.376003 

8.65 

8.63 
8.62 
8.60 

9.987496 
.987465 
.987434 
.987403 
.987372 

.52 
.52 
.52 
.52 

9.386438 
.386987 
.387536 

.388084 
.388631 

9.15 
9.15 
9.13 
9.12 

10.613562 
.613013 
.612464 
.611916 
.611369 

19 
18 
17 
16 
15 

46 
47 
48 
49 

.376519 
.377035 
.377549 
.378063 

8.60 
8.60 
8.57 
8.57 

.987341 
.987310 
.987279 
.987248 

.52 
.52 
.52 
.52 

fro 

.389178 
.389724 
.390270 
.390815 

9JO 
9.10 
9.08 
9  08 

.610822 
.610276 
.609730 
.609186 

14 
13 

12 
11 

50 

.378577 

8^53 

.987217 

.069 

.52 

.391360 

9^05 

.608640 

10 

51 

52 
53 
54 
55 
56 
57 
58 
59 
60 

9-379089 
.379601 
.380113 
.380624 
.381134 
.381643 
.382152 
.382661 
.383168 
9.383675 

8.53 
8.53 
8.52 
8.50 
8.48 
8.48 
8.48 
8.45 
8.45 

9.987186 
.987155 
.987124 
.987092 
.987061 
.987030 
.986998 
.986967 
.986936 
9.986904 

.52 
.52 
.53 
.52 
.52 
.53 
.52 
.52 
.53 

9.391903 
.392447 
.392989 
.393531 
.394073 
.394614 
.395154 
.395694 
.396233 
9.396771 

9.07 
9.03 
9.03 
9.03 
9.02 
9.00 
9.00 
8.98 
8.97 

10.608097 
.607553 
.607011 
.606469 
.605927 
.605386 
.604846 
.604306 
.603767 
10.603229 

9 
8 
7 
6 
5 

1 

2 
1 
0 

' 

Cosine. 

D.I". 

Sine. 

D.  1". 

!  Cotang. 

D.  r. 

Tang.  !  ' 

103° 


117 


14° 


COSINES,  TANGENTS,   AND  COTANGENTS. 


165° 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D:  r. 

Cotang. 

' 

0 

1 

2 

9.383675 

.384182 
.384687 

8.45 
8.42 

9.986904 

.986873 
.986841 

.52 
.53 

fcO 

9.396771 
.397309 
.397846 

8.97 
8.95 

8QC 

10.603229 
.602691 
.602154 

60 
59 

58 

3 

.385192 

o  'AC\ 

.986809 

.Do 

.398383 

.yo 

.601617 

57 

4 

5 

.385697 
.386201 

o.42 
8.40 

.986778 
.986746 

!53 

to 

.398919 
.399455 

8.  '93 

8  no 

.601081 
.600545 

56 
55 

6 

7 
8 

.386704 
.387207 
.387709 

8.38 
8.38 
8.37 

8  OK 

.986714 
.986683 
.986651 

.Do 

.52 
.53 

rq 

.399990 
.400524 
.401058 

.Ml 

8.90 
8.90 

800 

.600010 
.599476 
.598942 

54 
53 

52 

9 

.388210 

.OO 

.986619 

.Do 
to 

.401591 

.00 

8QQ 

.598409 

51 

10 

.388711 

8.35 
8.33 

.986587 

.Do 

.53 

.402124 

.OO 

8.87 

.597876 

50 

11 

9.389211 

9.986555 

tq 

9.402656 

80t 

10.597344 

49 

12 

.389711 

8.33 

.986523 

.Do 

53 

.403187 

.oD 

8  85 

.596813 

48 

13 

.390210 

Son 

.986491 

KO 

.403718 

8'pt 

.596282 

47 

14 
15 

.390708 
.391206 

.oU 

8.30 

.986459 
.986427 

.OO 

.53 

.404249 
.404778 

.  OO 

8.82 

8QQ 

.595751 
.595222 

46 
45 

16 

.391703 

8.28 

.986395 

.53 

.405308 

.OO 

.594692 

44 

17 

.392199 

8.27 

.986363 

.53 

.405836 

8.80 

8  on 

.594164 

43 

18 

.392695 

8.27 

.986331 

.53 

.406364 

.oU 

8  DA 

.593636 

42 

19 
20 

.393191 
.393685 

8.27 
8.23 
8.23 

.986299 
.986266 

.53 
.55 
.53 

.406892 
.407419 

.oU 

8.78 
8.77 

.593108 
.592581 

41 
40 

21 

9.394179 

9.986234 

to 

9.407945 

Q  w 

10.592055 

39 

22 
23 

.394673 
.395166 

8.22 

Son 

.986202 
.986169 

.Do 

.55 

to 

.408471 
.408996 

8i75 

87K 

.591529 
.591004 

38 
37 

24 

.395658 

.fi\) 

.986137 

.DO 

.409521 

.  ID 

.590479 

36 

25 
26 

.396150 
.396641 

8.20 
8.18 
81ft 

.986104 
.986072 

.55 
.53 
55 

.410045 
.410569 

8.'73 

8  72 

.589955 
.589431 

35 
34 

27 

.397132 

.  lo 
81  ^ 

.986039 

.411092 

8'r-p 

.588908 

33 

28 

.397621 

.  ID 

81  "• 

.986007 

KK 

.411615 

.  I  .* 
8M 

.588385 

32 

29 
30 

.398111 
.398600 

.  li 

8.15 
8.13 

.985974 
.985942 

.55 
.53 
.55 

.412137 
.412658 

.  i  U 

8.68 
8.68 

.587863 
.587342 

31 
30 

31 
32 
33 
34 

9.399088 
.399575 
.400062 
.400549 

8.12 
8.12 
8-.  12 

9.985909 

.985876 
.985843 
.985811 

.55 
.55 
.53 

9.413179 
.413699 
.414219 
.414738 

8.67 
8.67 
8.65 

10.586821 
.586301 

.585781 
.585262 

29 
28 
27 
26 

35 

.401035 

8.  10 

8  no 

.985778 

.55 

tt 

.415257 

SCO 

.584743 

25 

36 

.401520 

.Uo 

.985745 

.DD 

.415775 

.  OO 

.584225 

24 

37 

.402005 

8.08 

.985712 

.5") 

re 

.416293 

2*2 

.583707 

23 

38 

.402489 

8.07 

8f\K. 

.985679 

.DO 

.416810 

8  fin 

.583190 

22. 

39 
40 

.40297'2 
.403455 

.Uo 
8.05 
8.05 

.985646 
.985613 

.55 

.55 
.55 

.417326 
.417842 

.OU 

8.60 
8.60 

.582674 

.582158 

21 
20 

41 
42 
43 
44 
45 
46 

9.403938 
.404420 
.404901 
.405382 
.405862 
.406341 

8.03 
8.02 
8.02 
8.00 
7.98 

9.985580 
.985547 
.985514 
.985480 
.985447 
.985414 

.55 
.55 
.57 
.55 
.55 

9.418358 
.418873 
.419387 
.419901 
.420415 
.420927 

8.58 
8.57 
8.57 
8.57 
8.55 

Q  tt 

10.581642 
.581127 
.580613 
.580099 
.579585 
.579073 

19 
18 
17 
16 
15 
14 

47 

.406820 

".98 

.985381 

.  55 

.421440 

o.DD 
8  to 

.578560 

13 

48 
49 

.407299 
.407777 

".98 

r/97 

.985347 
.985314 

.57 
.55 
57 

.421952 
.422463 

.  Do 

8.52 
8  5*-* 

.578048 
.577537 

12 
11 

50 

.408254 

^95 

.985280 

'.55 

.422974 

8.'50 

.577026 

10 

51 
52 

9.408731 

.409207 

".93 

9.985247 
.985213 

.57 

9.423484 
.423993 

8.48 

8KA 

10.576516 

.576007 

9 

8 

53 
54 

.409682 
.410157 

.93 

.92 

.985180 
.985146 

.55 
.57 

.424503 
.425011 

.  OU 

8.47 

8Aiy 

.575497 
.574989 

7 
6 

55 

.410632 

.92 

.985113 

.55 

.425519 

.4i 

.574481 

5 

56 

.411106 

.90 

00 

.985079 

ten 

.426027 

8  45 

.573973 

4 

57 

.411579 

.  OO 

.985045 

.57 

.426534 

.573466 

3 

58 
59 

.412052 
.412524 

.88 

.87 

07 

.985011 

.984978 

!55 

.427041 
.427547 

8^43 
.4~ 

.572959 
.572453 

2 
1 

60  I 

9.412996 

.01 

9.984944 

.57 

9.428052 

10.571948 

0 

/  R 
E 

Cosine. 

D.  r. 

Sine. 

D.  r. 

Cotang. 

D.  1'. 

Tang.  | 

' 

104° 


118 


75C 


15° 


TABLE    X. — LOGARITHMIC    SINES, 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

9.412996 
.413467 

7.85 

9.984944 
.984910 

.57 

C*1 

9.428052 
.428558 

8.43 

10.571948 
.571442 

60 
59 

2 
3 

.4139138 
.414408 

7.85 
7.83 

r>  oq 

.984876 
.984842 

.o< 
.57 

e-7 

.429062 
.429566 

8*40 

.570938 
.570434 

58 
57 

4 

.414878 

t  .00 

.984808 

,94 

.430070 

o.4U 

.569930 

56 

5 
6 

7 

.415347 
.415815 
.416283 

7.82 
7.80 
7.80 

.984774 
.984740 
.984706 

.57 

.57 
.57 

.430573 
.431075 
.431577 

8.38 
8.37 
8.37 

.569427 
.568925 
.568423 

55 
54 
53 

8  t  .416751 

7.80 

.984672 

.57 

.432079 

8.37 

.567921 

52 

9 
10 

.417217 
.417684 

7.77 
7.78 

7.77 

.984638 
.984603 

.57 
.58 
.57 

.432580 
.433080 

8.35 
8.33 
8.33 

.567420 
.566920 

51 
50 

11 

9.418150 

9.984569 

9.433580 

10.566420 

49 

12 
13 
14 

.418615 
.419079 
.419544 

7.75 
7.73 
7.75 

.984535 
.984500 
.984466 

!ss 

.57 

.434080 
.434579 

.435078 

8.o3 
8.32 
8.32 

.565920 
.565421 
.564922 

48 
47 
46 

15 
16 

.420007 
.420470 

7.72 
7.72 

.984432 
.984397 

.57 

.58 

.435576 
.436073 

8.30 

8.28 

.564424 
.563927 

45 
44 

17 
18 

.420933 
.421395 

7.72 
7.70 

.984:363 
.984328 

.57 
.58 

.436570 
.437067 

8.28 
8.28 

.563430 
.562933 

43 
42 

19 
20 

.421857 
.422318 

7.70 
7.68 
7.67 

.984294 
.984259 

.57 

.58 
.58 

.437563 
.438059 

8.27 
8.27 
8.25 

.562437 
.561941 

41 

40 

21 

3.422778 

9.984224 

9.438554 

10.561446 

39 

22 

.423288 

7.67 

.984190 

.57 

KO 

.439048 

8.~o 

.560952 

38 

23 

.423697 

7.65 

.984155 

..JO 
RQ 

.439543 

8./&5 

.560457 

37 

24 
25 

.424156 
.424615 

7.65 
7.65 

.984120 
.984085 

.OO 

.58 

fO 

.440036 
.440529 

8.22 
8.22 

8  no 

.559964 
.559471 

36 
35 

26 
27 
28 
29 

.425073 
.425530 
425987 
.426443 

7.63 
7.62 
7.62 
7.60 

.984050 
.984015 
.983981 
.983946 

.00 
.58 
.57 

.58 

to 

.441022 
.441514 
.442006 
.442497 

.64 

8.20 
8.20 
8.18 

810 

.558978 
.558486 
.557994 
.557503 

34 
33 
32 
31 

30 

.426899 

7.60 

.983911 

.Do 

.442988 

.  lo 

.557012 

30 

7.58 

.60 

8.18 

31 

9.427354 

9.983875 

to 

9.443479 

81^ 

10.556521 

29 

32 

.427809 

7.58 

.983840 

.DO 

.443968 

.  lo 

.556032 

28 

33 

.428263 

7.57 

.983805 

.58 

.444458 

8.17 

.555542 

27 

34 
35 

.428717 
.429170 

7.57 
7.55 

.983770 
.983735 

.58 

.58 

.444947 
.445435 

8.15 
8.13 

.555053 
.554565 

26 
25 

36 

.429623 

7.55 

.983700 

.58 
fin 

.445923 

8.13 

8-jO 

.554077 

24 

37 

.430075 

7  .  53 

.9&S664 

.ou 

.446411 

.  lo 

.553589 

23 

38 
39 
40 

.430527 
.430978 
.431429 

7  53 
7.52 
7.52 

,9aS629 
.983594 
.983558 

.58 
.58 
.60 

.446898 
.447384 
.447870 

8.12 
8.10 
8.10 

.553102 
.552616 
.552130 

22 
21 
20 

7.50 

.58 

8.10 

41 

9.431879 

9.983523 

9.448356 

o  no 

10.551644 

19 

42 
43 

.432329 
.432778 

7.50 

7.48 

.983487 
.983452 

.60 

.58 

.448841 
.449326 

o.Oo 
8.08 

.551159 
.550674 

18 
17 

44 
45 
46 

.4&3226 
.433675 
.434122 

7.47 
7.48 
7.45 

.983416 
.983381 
.983345 

.60 
.58 
.60 

.449810 
.450294 
.450777 

8.07 
8.07 
8.05 

.550190 
.549706 
.549223 

16 
15 
14 

47 

48 

.434569 
.435016 

7.45 
7.45 

.983309 
.983273 

.60 
.60 

KQ 

.451260 
.451743 

8.05 
8.05 

8rvo 

.548740 
.548257 

13 

12 

49 

.435462 

7.43 

.983238 

.00 

.452225 

.Uo 

.  547775 

11 

50 

.435908 

7.43 

7.42 

.983202 

.60 
.60 

.452706 

8.02 
8.02 

.547294 

10 

51 

9.436353 

9.983166 

9.453187 

8  no 

10.546813 

9 

52 
53 
54 

.436798 
.437242 
.437686 

7.42 
7.40 
7.40 

.983130 
.983094 
.983058 

.60 
.60 
.60 

.453668 
.454148 
.454628 

.  \}6 

8.00 
8.00 

r>  QQ 

.546332 
.545852 
.545372 

8 
7 
6 

55 

.438129 

7.38 

.983022 

.60 

.455107 

i  .yo 

.544893 

5 

56 
57 

.438572 
.439014 

7.38 
7.37 

.982986 
.982950 

.60 
.60 

.455586 
.456064 

7.98 
7.97 

.544414 
.543936 

4 
3 

58 

.439456 

7.37 

.982914 

.60 

.456542 

7.97 

.543458 

2 

59   .439897 

7.35 

.982878 

.60 

.457019 

7.95 

7QC 

.542981 

1 

60  9.440338 

7.35 

9.982842 

.60 

9.457496 

.yo 

10.542504 

0 

'   Cosine. 

D.  r.  I 

Sine.  |  D.  1".  i  Cotang.  D.  1'. 

Tang. 

' 

105° 


119 


COSINES,   TANGENTS,   AND  COTANGENTS. 


, 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

, 

i 

0 

1 

2 

9.440338 

.440778 
.441218 

7.33 
7.33 

r*  qq 

9.982842 
.982805 
.982769 

.62 
.60 
fift 

9.457496 
.457973 

.458449 

7.95 

7.93 

10.542504 
.542027 
.541551 

60 

59 

58 

3 

4 

.441658 
.442096 

t  .OO 

£-30 

.982733 
.982696 

.ou 
.62 
fin 

.458925 
.459400 

7  '.92 

7  Q9 

.541075 
.540600 

57 
56 

5 

.442S&5 

7  'Vl 

.982660 

.ou 
60 

.459875 

i  .  .U. 

7  90 

.540125 

55 

6 

7 

.442973 
.443410 

I  .OU 

7.28 

.982624 

.982587 

!62 

.460349 
.460823 

7J90 

.539651 
.539177 

54 

53 

8 
9 
10 

.443847 
.444284 
.444720 

7.28 
7.28 
7.27 
7.25 

.982551 
.982514 
.982477 

.60 
.62 
.62 
.60 

.461297 
.461770 
.462242 

^SS 
7.87 
7.88 

.538703 
.538230 
.537758 

52 
51 
50 

11 

9.445155 

7  9** 

9.982441 

62 

9.462715 

7  85 

10.537285 

49 

12 

.445590 

i  ,XSt 

.982404 

.463186 

(-,'07 

.536814 

48 

13 
14 

.446025 
.446459 

7  '.23 

.982367 
.982331 

!eo 

.463658 
.464128 

<  .01 

7.83 

.536342 
.535872 

47 
46 

15 

.446893 

r-'oo 

.982294 

62 

.464599 

7  83 

.535401 

45 

16 

.447326 

n  oo 

.982257 

.465069 

7oq 

.534931 

44 

17 

.447759 

i  ,TBt 

.982220 

62 

.465539 

.OO 

7  82 

.534461 

43 

18 

.448191 

•~  9ft 

.982183 

.466008 

.533992 

42 

19 

.448623 

"  1  ft 

.982146 

62 

.466477 

7  80 

.533523 

41 

20 

.449054 

7!l8 

.982109 

.466945 

7^80 

.533055 

40 

21 
22 
23 
24 

9.449485 
.449915 
.450345 
.450775 

7.17 
7.17 
7.17 

7  -j  K 

9.982072 
.982035 
.981998 
.981961 

.62 
.62 
.62 
62 

9.467413 

.467880 
.468347 
.468814 

7.78 
7.78 
7.78 

10.532587 
.532120 
.531653 
.531186 

39 

38 
37 
36 

25 

.451204 

<  .  JO 

.981924 

"co 

.469280 

7  '77 

.530720 

35 

26 

.451632 

it  1Q 

.981886 

.00 

.469746 

i  .  1  1 

.530254 

34 

27 

28 

.452060 

.452488 

7.  lo 
7.13 

f"  19 

.981849 
.981812 

!62 

.470211 
.470676 

?-£5 

.529789 
.529324 

33 

32 

29 
30 

.452915 
.453342 

<  .  \.4i 

7.12 
7.10 

.981774 
.981737 

!62 
.62 

.471141 
.471605 

7^73 
7.73 

.528859 
.528395 

31 

30 

31 

9.453768 

^  in 

9.981700 

on 

9.472069 

7  72 

10.527931 

29 

32 

.454194 

t  .  JU 

.981662 

.Oo 

.472532 

7  72 

527468 

28 

33 

.454619 

'  'Ho 

.981625 

nn 

.472995 

.527005 

27 

34 
35 
36 

.455044 
.455469 
.455893 

7.08 
7.08 
7.07 

7  AK 

.981587 
.981549 
.981512 

.63 

.63 

.62 

on 

.473457 
.473919 
.474381 

7!  70 
7.70 

7  RP, 

.526543 
.526081 
.525619 

26 
25 
24 

37 

.456316 

(  .  uo 

.981474 

.Oo 

.474842 

l  .  OO 
r>  CO 

.525158 

23 

38 
39 
40 

.456739 
.457162 
.457584 

7.05 
7.05 
7.03 
7.03 

.981436 
.981399 
.981361 

.63 

.62 
.63 
.63 

.475303 
.475763 
.476223 

t  .Oo 

7.67 
7.67 
7.67 

.524697 
.524237 

.523777 

22 
21 
20 

41 
42 
43 
44 
45 

9.458006 

.458427 
.458848 
.459268 
.459688 

7.02 
7.02 
7.00 
7.00 

9.981323 
.981285 
.981247 
.981209 
.981171 

.63 
.63 
.63 
.63 

9.476683 
.477142 
.477601 
.478059 
.478517 

7.65 
7.65 
7.63 

£-63 

10.523317 
.522858 
.522399 
.521941 
.521483 

19 
18 
17 
16 
15 

46 

.460108 

7.00 

6QQ 

.981133 

.63 

on 

.478975 

7  62 

.521025 

14 

47 

.460527 

.  yo 

6QQ 

.981095 

.00 
63 

.479432 

7  62 

.520568 

13 

48 

.460946 

.yo 

.981057 

.479889 

i~  fin 

.520111 

12 

49 
50 

.461364 
.461782 

6.97 
6.97 
6.95 

.981019 
.980981 

.63 
.63 
.65 

.480345 

.480801 

4  .OU 

7.60 
7.60 

.519655 
.519199 

11 
10 

51 

9.462199 

9.980942 

9.481257 

7  58 

10.518743 

9 

52 
53 
54 
55 
56 

.462616 
.463032 
.463448 
.463864 
.464279 

6.95 
6.93 
6.93 
6.93 
6.92 

.980904 
.980866 
.980827 
.980789 
.980750 

!63 
.65 
.63 
.65 

on 

.481712 
.482167 
.482621 
.483075 
.483529 

7  '.57 
7.57 
7.57 
7  55 

.518288 
.517833 
.517379 
.516925 
.516471 

8 

6 
5 
4 

57 
58 
59 
60 

.464694 
.465108 
.465522 
9.465935 

6.92 
6.90 
6.90 
6.88 

.980712 
.980673 
.980635 
9.980596 

.Oo 

.65 
.63 
.65 

.483982 
.484435 
.484887 
9.485339 

7i  55 
7.53 
7.53 

.516018 
.515565 
.515113 
10.514661 

3 

2 
1 
0 

' 

Cosine. 

D.  1". 

Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

' 

120 


17° 


TABLE    X. — LOGARITHMIC    SINES, 


' 

Sine. 

D.  r. 

Cosine. 

D.  r. 

Tang. 

D.  1". 

Cotang. 

f 

0 

9.465935 

600 

9.980596 

AO. 

9.485339 

7KO 

10.514661 

60 

1 
8 

.466348 
.466761 

.00 

6.88 

607  ' 

.<KS0558 
.980519 

.Do 

.65 
A*; 

.485791 
.486242 

.OO 

7.52 

7K9 

.514209 
.513758 

59 

58 

3 

4 
5 

.467173 
.467585 
.467996 

.at   \ 

6.87 
6.85 

6  OK 

.980480 
.980442 
.980403 

.DO 

.63 
.65  | 

.486693 
.487143 
.487593 

.(KB 

7.50 
7.50 

.513307 
.512857 
.512407 

57 
56 
55 

6 

8 
9 
10 

.468407 
.468817 
.469227 
.469637 
.470046 

.OO 

6.83 
6.83 
6.83 
6.82 
6.82 

.980:364 
.980325 
.980286 
.980247 
.980208 

'.65 
.65 
.65 
.65 
.65 

.488043 
.488492 
.488941 
.489390 
.489838 

7^48 
7.48 
7.48 
7.47 

7.47 

.511957 
.511508 
.511059 
.510610 
.510162 

54 
53 
52 
51 
50 

11 
12 
13 

9.470455 

.470863 
.471271 

6.80 
6.80 

9.980169 
.980130 
.980091 

.65 
.65 

9.490286 
.4iK)733 
.491180 

7.45 
7.45 

10.509714 
.509267 
.508820 

49 
48 
47 

14 

.471679 

6.80 

.980052 

.65 

.491627 

7.45 

.508373 

46 

15 
16 

.472086 
'.472492 

D.  iO  ', 
6.77 

677 

.980012 
.979973 

.07 
.65 

CK 

.492073 
.492519 

7.43 
7.43 

.507927 
.507481 

45 
44 

17 

.472898 

.  t  { 

677 

.979934 

.DO 
OK 

.492965 

'•'49 

.507035 

43 

18 

.473304 

.  ff 

.979895 

.  DO 

.493410 

r-'*f\ 

.506590 

42 

19 

.473710 

6.77 

.979855 

.67 

pe 

.493854 

t  .40 

749 

.506146 

41 

20 

.474115 

6>3 

.979816 

.DO 

.67 

.494299 

.4/4 

7.40 

.505701 

40 

21 
22 
23 

9.474519 
.474923 
.475327 

6.73 
6.73 

679 

9.979776 
.979737 
.979697 

.65 

.67 

9.494743 
.495186 
.495630 

7.38 
7.40 

10.505257 
.504814 
.504370 

39 

38 
371 

24 

.475730 

.  <  ~ 

.979658 

.65 

.496073 

'  'oS 

.503927 

36 

25 

.476133 

6.72  i 

679   ! 

.979618 

.67 

CK 

.496515 

7.o7 

7  ^7 

.503485 

35 

26 
27 

28 

.476536 
.476938 
.477340 

.  <~ 

6.70 
6.70 

6  Aft   ' 

.979579 
.979539 
.979499 

.DO 

.67 
.67 

.496957 
.497399 
.497841 

1  .04 

7.37 
7.37 

7  '-^^ 

.503043 
.502601 
.502159 

34 
33 
32 

29 

.477741 

.Do   ; 

.979459 

.67 

.498282 

<  .  oO 

.501718 

31 

30 

.478142 

6.68 
6.67  j 

.979420 

.65 
.67 

.498722 

7  .  33 
7.35 

.501278 

30 

31 
32 
33 

9.478542 

.478942 
.479342 

6.67 
6.67 

9.979380 
.979:340 
.979300 

.67 
.67 

9.499163 
.499603 
.500042 

7.as 

7.32 

10.500837 
.500397 
.499958 

29 

28 
27 

34 

.479741 

6.65 

.979260 

.67 

.500481 

7.32 

n  09 

.499519 

26 

35 
36 

.480140 
.480539 

6.65 
6.65 

.979220 
.979180 

.67 
.67 

A7 

.500920 
.501359 

4  .06 

7.32 

7  30 

.499080 
.498641 

25 
24 

37 

38 
39 
40 

.480937 
.481334 
.481731 
.482128 

6^62  ! 
6.62 
6.62  ! 
6.62 

.979140 
.979100 
.979059 
.979019 

.O< 

.67 
.68 
.67 
.67 

.501797 
.502235 
.502672 
.503109 

7^30 

7.28 
7.28 
7.28 

.498203 
.497765 
.497328 
.496891 

23 
22 
21 

20 

41 

9.482525 

9.978979 

9.503546 

10.496454 

19 

42 

1  .482921 

6.60 

.978939 

.67 

.503982 

7.27 

.496018 

18 

43 

.483316 

6.58 

6  Aft 

.978898 

.68 

.504418 

7.27 

n  97 

.495582 

17 

44 

.483712 

.  DU 

.978858 

r> 

.504854 

i  .«< 

.495146 

16 

45 

.484107 

6.58 

.978817 

.68 

.505289 

7.25 

.494711 

15 

46 

.484501 

O.O< 

.978777 

.67 

.505724 

7.25 

.494276 

14 

47 

.484895 

6.57 

6K.7 

.978737 

.67  j 

Aft 

.506159 

7.25 

7  9°. 

.493841 

13 

48 

.485289 

.Ol 

6KK 

.978606 

.Do   | 
Aft 

.506593 

1  .&) 
790 

.493407 

12 

49 

.485682 

.OO 

.978655 

.Do 

.507027 

.  Mi 

.492973 

11 

50 

.486075 

6.55 
6.53 

.978615 

.67 

.68 

.507460 

7.23 
7.22 

.492540 

10 

51 

19.486467 

6KK 

9.978574 

Aft 

9.507893 

P»  09 

10.492107 

9 

52 
53 

F  .486860 

.487251 

.DO 

6.52 

6  CO 

.978533 
.978493 

.Do   ! 

.67 

Aft 

.508326 
.508759 

7^22 

7  9O 

.491674 
.491241 

8 

7 

54 

.487643 

.Oo 
R  K9 

.978452 

.Oo 
Aft 

.509191 

t  ,<M 

71ft 

.490809 

6 

55 

.488034 

D.O.O 

.978411 

.Do   > 

.509622 

.  lo 

.490378 

5 

56 

.488424 

6.50 
6  en 

.978370 

.68 

Aft 

.510054 

7  1ft 

.489946 

4 

57 

.488814 

.  OU 

978329 

.Oo 

!  .510485 

t  .  lo 

.489515 

3 

58 

.489204 

6.50 

.978288 

.68 

i  .510916 

7.18 

.489084 

2 

59 

.489593 

6.48 

.978247 

.68 

i  .511346 

7.17 

.488654 

1 

60 

9.489982 

6.48 

9.978206 

.68 

9.511776 

7.17 

10.488224 

0 

i 

Cosine. 

D  r. 

Sine. 

D.  r. 

Cotang. 

D.  1'. 

Tang. 

107 


121 


COSINES,   TANGENTS,  AND   COTANGENTS. 


161° 


' 

Sine. 

D.  1". 

Cosine. 

ll       ' 
D.  1".    Tang. 

D.  1". 

Cotang. 

' 

0  !  9.489982 
1  !  .490371 

6.48 

9.97'8206 
.978165 

.68 
Aft 

9.511776 
.512206 

7.17 

10.488224 

.487794 

60  : 
59 

2  !  .490759 

6.47 

647 

.978124 

.Do 
fift 

.512635 

7.15 

i  .  15 

.487365 

58 

3  !  .491147 
4  i  .491535 

.4< 

6.47 

.978083 
.978042 

.  Do 

.68 
fift 

.513064 
.513493 

7.15 

71°. 

.486936 
.486507 

57 

56 

5   .491922 

^   !  .978001 

.Do 

.513921 

•  .  lo 
71°. 

.486079 

55 

6   .492308 

O.4O   | 

.977959 

'  Ao 

.514349 

.  Jo 

.485651 

54 

7   .492695 

6.45 

.977918 

.DO 

.514777 

7.13 

719 

.485223 

53 

8   .493081 

6.43 

.977877 

•25 

.515204 

*  .  IX; 

.484796 

52 

9 
10 

.493466 
.493851 

6.42 
6.42 
6.42 

.977835 
.977794 

.  <o 

.68 
.70 

.515631 
.516057 

7.12 
7.10 

7.12 

.484369 
.483943 

51 
50 

11 
12 
13 

9.494236 
.494621 
.495005 

6.42 
6.40 
600 

9.977752 
.977711 
.977669 

.68 
.70 
fift 

9.516484 
.516910 
.517335 

7.10 
7.08 
7  10 

10.483516 
.483090 
.482665 

49 

48 

47 

14 

.495388 

.00 

.977628 

.Do 
r'O 

.517761 

.482239 

46 

15 

.495772 

60,7 

.977586 

.  tO 

70 

.518186 

7  07 

.481814 

45 

16 

.496154 

.04 

6OQ 

.977544 

.  t  U 

fift 

.518610 

7  07 

.481390 

44 

17 

.496537 

.OO 

.977503 

.Do 

.519034 

i  .U/ 

.480966 

43 

18 

.496919 

6.37 

.977461 

.70 

.519458 

i  .07 

.480542 

42 

19 

.497301 

6.37 

.977419 

.70 

.519882 

i  .07 

7  HK. 

.480118 

41 

20 

.497682 

6.  35 
6.35 

.977377 

.70 
.70 

.520305 

t  .Uo 
7.05 

.479695 

40 

21 
22 

9-498064 
.498444 

6.33 

6  OK 

9.977335 
.977293 

•1Q 

9.520728 
.521151 

7.05 
70°. 

10.479272 

.478849 

39 

38 

23 

.498825 

.OO 

.977251 

jJJ 

.521573 

.UO 

.478427 

37 

24 
25 

.499204 
.499584 

6.32 
6.33 

6  op 

.977209 
.977167 

.  (0 
.70 

"~0 

.521995 
.522417 

7^03 

7  02 

.478005 

.477583 

36 
35 

26 

.499963 

.O(C 

.977125 

.  <  u 

.522838 

.477162 

34 

27 
28 

.500342 
.500721 

6^32 
60,0 

.977083 
.977041 

!70 

mn 

.523259 
I  .523680 

7  '.02 

.476741 
.476320 

33 
32 

29 

30 

.501099 
.501476 

.oU 

6.28 
6.30 

.976999 
.976957 

.  <u 
.70 

72 

.524100 
.524520 

7^00 
7.00 

.475900 
.475480 

31 
30 

31 

9.501854 

9.976914 

9.524940 

6  Oft 

10.475060 

29 

32 

.502231 

ft  O7 

.976872 

An 

.525359 

.  »o 
60ft 

.474641 

28 

33 

.502607 

69ft 

.976830 

»2S 

.525778 

.  »o 
6  Oft 

.474222 

27 

34 

.502984 

.(CO 

.976787 

.  7* 

.526197 

.  Jo 

.473803 

26 

35 
36 
37 

38 

.503360 
.503735 
.504110 
.504485 

6.27 
6.25 
6.25 
6.25 

.976745 
.976702 
.976660 
.976617 

.70 

.72 
.70 

.72 

.526615 
.527033 
.527451 

.527868 

6.97 
6.97 
6.97 
6.95 

.473385 
.472967 
.472549 
.472132 

25 
24 
23 
22 

39 

.504860 

6.25 

.976574 

.72 
•Tin 

.528285 

6.95 
6  95 

.471715 

21 

40 

.505234 

6^23 

.976532 

.  <u 

.72 

.528702 

6^95 

.471298 

20 

41 
42 
43 

9.505608 
.505981 
.506354 

6.22 
6.22 

9.976489 
.976446 
.976404 

.72 

•r-° 

9.529119 
.529535 
.529951 

6.93 
6.93 
6  92 

10.470881 
.470465 
.470049 

19 

18 
17 

44 

.506727 

?\ 

.976361 

.  <2 

.530366 

.469634 

16 

45 

.507099 

6.20 

.976318 

.72 

.530781 

D.  JA 

.469219 

15 

46 

47 
48 

.507471 

.507843 
.508214 

6.20 
6.20 
6.18 
61ft 

.976275 
.976232 
.976189 

.7'2 
.72 
.72 

r-o 

.531196 
.531611 
.532025 

6^92 
6.90 

.468804 
.468389 
.467975 

14 
13 
12 

49 
50 

.508585 
.508956 

.  Jo 

6.18 
6.17 

.976146 
.976103 

>2 
72 

.532439 
.532853 

6  '.90 

6.88 

.467561 
.467147 

11 
10 

51 

52 

9.509326 
.509696 

6.17 

9.976060 
.976017 

.72 

9.533266 
.533679 

6.88 

10.466734 
.466321 

9 

8 

53 

.510065 

6.15 
61  K. 

.975974 

.  C6 

.534092 

6.88 

fi  87 

.465908 

7 

54 

.510434 

.  it) 

.975930 

•22 

.534504 

U.oi 

Grtrt 

.465496 

6 

55 

56 

.510803 
.511172 

6.15 
6.15 

.975887 
.97'5844 

.  <2 

.72 

.534916 

.535328 

.  Ol 

6.87 

6  OK 

.465084 
.464672 

5 
4 

57 

.511540 

6.  13 

.975800 

.73 

.535739 

.  oO 

.464261 

3 

58 

.511907 

6.12 

.975757 

'L 

.536150 

6.85 

.463850 

2 

59 
GO 

.512275 
9,512642 

6.13 
6.13 

.975714 
9.975670 

.72 
.73 

.536561 
9.530972 

6.85 
6.85 

.463439 
10.463028 

1 

0 

' 

C'osino. 

I).  1". 

Sine. 

D.r. 

Cotang.  D.  1".   Tang.    ' 

108» 


71° 


19° 


TABLE    X. — LOGARITHMIC    SINES, 


1GO« 


' 

Sine. 

D.  r. 

Cosine. 

D.  r. 

Tang. 

D.  1'. 

Cotang. 

' 

o 

1 

9.512642 
.513009 

6.12 

9.975670 
.975627 

.72 

9.536972 
.537382 

6.83 

10.463028 
.462618 

60 
59 

2 

.513:375 

6.10 

.975583 

.73 

.537792 

6.83 

.462208 

58 

3  1  1513741 
4  1  .51410? 

6.10 
6.10 

.975539 
.975496 

.73 
.72 

.538202 
.538611 

6.83 
6.82 

.461798 
.461389 

57 
56 

5 

6 

.514472 
.514837 

6.08 
6.08 

.975452 
.975408 

.73 
.73 

.539020 
.539429 

6.82 
6.82 

.460980 
.460571 

55 
54 

.515202 

6.08 

C  r\7 

.975365 

*s 

.539837 

6.80 

6  Of) 

.460163 

53 

S 
9 
10 

.515566 
.5l59tt) 
.516294 

O.v'< 

6.07 
6.07 
6.05 

.975321 
.975277 
.975233 

'.73 
.73 
.73 

.540245 
.540653 
.541061 

.ou 
6.80 
6.80 
6.78 

.459755 
.459347 
.458939 

52 
51 

50 

11 
12 
13 
14 
15 
16 

9.516657 
.517020 
.517382 
.517745 
.518107 
.518468 

6.05 
6.03 
6.05 
6.03 
6.02 

9.975189 
.975145 
.975101 
.975057 
.975013 
.974969 

.73 
.73 
.73 
.73 
.73 

9.541468 
.541875 
.542281 
.542688 
.543094 
.543499 

6.78 
6.77 
6.78 
6.77 
6.75 

10.458532 
.458125 
.457719 
.457312 
.456906 
.456501 

49 

48 
47 
46 
45 
44 

17 

.518829 

6.02 

.974925 

.73 

.543905 

6.77 

.456095 

43 

18 

.519190 

6.02 

.974880 

.75 

.544310 

6.75 

.455690 

42 

19 
20 

.519551 
.519911 

6.02 
6.00 
6.00 

.974836 
.974792 

.73 
.73 
.73 

.544715 
.545119 

6.75 
6.73 
6.75 

.455285 

.454881 

41 
40 

21 
22 

9.520271 
.520631 

6.00 

9.974748 
.974703 

.75 

9.545524 

.545928 

6.73 

10.454476 
.454072 

39 

38 

23 

.520990 

5.98 

.974659 

.73 

.546331 

6.72 

,453669 

37 

24 
25 
26 
27 

28 
29 

.521349 
.521707 
.522066 
.522424 
.522781 
.523138 

5.98 
5.97 
5.98 
5.97 
5.95 
5.95 

.974614 
.974570 
.974525 
.974481 
.974436 
.974391 

.75 
.73 
.75 
.73 
.75 
.75 

.546735 
.547138 
.547540 
.547943 
.548345 
.548747 

6.73 
6.72 
6.70 
6.72 
6.70 
6.70 

.453265 
.452862 
.452460 
.452057 
.451655 
.451253 

36 
35 
34 
33 
32 
31 

30 

.523495 

5.95 

.974347 

.73 

.549149 

6.70 

.450851 

30 

5.95 

.75 

6.68 

31 
32 

9.523852 
.524208 

5.93 

9.974302 
.974257 

.75 

9.549550 
.549951 

6.68 

10.450450 
.450049 

29 

28 

33 

.524564 

5.93 

.974212 

.75 

.550352 

6.68 

.449648 

27 

34 

.524920 

5.93 

.974167 

.75 

.550752 

6.67 

.449248 

26 

35 

.525275 

5.92 

.974122 

.75 

.551153 

6.68 

.448847 

25 

36 
37 

.525630 

.525984 

5.92 
5.90 

.974077 
.974032 

.75 
.75 

.551552 
.551952 

6.65 
6.67 

.448448 
.448048 

24 
23 

38 
39 
40 

.526339 
.526693 
.527046 

5.92 
5.90 
5.88 
5.90 

.973987 
.973942 
.973897 

.75 
.75 
.75 
.75 

.552351 
.552750 
.553149 

6  .  65 
6.65 
6.65 
6.65 

.447649 
.447250 
.446851 

22 
21 
20 

41 

42 
43 
44 
45 

46 

9.527400 
.527753 

.528105 
.528458 
.528810 
.529161 

5.88 
5.87 
5.88 
5.87 
5.85 

9.973852 
.973807 
.973761 
.973716 
.973671 
.973625 

.75 

.77 
.75 
.75 

.77 

9.553548 
.553946 
.554344 
.554741 
.555139 
.55^536 

6.63 
6.63 
6.62 
6.63 
6.62 

10.446452 
.446054 
.445656 
.445259 
.444861 
.444464 

19 
18 
17 
16 
15 
14 

47 

.529513 

5.87 

.973580 

.75 

.555933 

6.62 

.444067 

13 

48 

.529864 

5.85 

.973535 

.75 

.556329 

6.60 

.443671 

12 

49 
50 

.530215 
.530565 

5.85 
5  ".83 
5.83 

.973489 
.973444 

.75 

.556725 
.557121 

6.60 
6.60 
6.60 

.443275 

.442879 

11 
10 

51 
52 

9.530915 
.531265 

5.83 

9.973398 
.973352 

.77 

9.557517 
.557913 

6.60 

10.442483 
.442087 

9 

8 

53 

.531614 

5.82 

.97.3307 

.75 

.558808 

6  .  58 

.441692 

7 

54 

.531963 

5.82 

.973261 

•  *,* 

.558703 

6  .58 

.441297 

6 

55 

.532312 

5.82 

.973215 

•  '  ( 

.559097 

6.57 

.440903 

5 

56 

57 
58 

.532661 
.533009 
.533357 

5.82 
5.80 
5.80 

.973169 
.973124 
.973078 

.77 
•£5 

.559491 
.559885 
.560279 

6^57 
6.57 

6tr< 

.440509 
.440115 
.439721 

4 
3 
2 

59 
60 

.533704 
9.534052 

5.78 
5.80 

.973032 
9.972986 

!77 

.560673 
9.561066 

.O< 

6.55 

.439327 
10.438934 

1 
0 

'   Cosine. 

i>.  r. 

1  Sine.   D.  1". 

Cotang. 

I).  1". 

Tang. 

' 

109* 


123 


COSINES,  TANGENTS,  AND  COTANGENTS. 


159* 


> 

Sine. 

D.  r. 

Cosine. 

D.  r. 

Tang. 

D.  r. 

Cotang. 

> 

0 

1 

2 
3 

9.534052 
534399 
.534745 
.535092 

5.78 
5.77 

5.78 

9.972986 
.972940 
.972894 
972848 

.77 
.77 
77 

77 

9  561066 
.561459 
561851 
562244 

6  55 
6  53 
6.55 
6  53 

10.438934 
.438541 
.438149 
.437756 

60 

59 
58 
57 

4 

.535438 

5.77 

972802 

i  t 

7ft 

.562636 

.437364 

56 

5 
6 

8 

535783 
.536129 
.536474 
:  .536818 

5.75 
5.77 
5  75 
5.73 

972755 
.972709 
.972663 
972617 

to 

.77 
.77 
77 

7ft 

.563028 
563419 
563811 
.564202 

6  52 
6  53 
6  52 
6  52 

.436972 
.436581 
.436180 
.435798 

55 
54 
53 

52 

9 
10 

.537163 
.537507 

5.75 
5.73 
5.73 

972570 
.972524 

to 

,77 
.77 

.564593 
.564983 

6  50 
6.50 

.435407 
.435017 

51 
50 

11 
12 
13 
14 
15 
16 

9.537851 
538194 
.538538 
.538880 
.539223 
539565 

5.72 

5.73 
5.70 
5.72 
5.70 

9.972478 
972431 
972385 
972338 
.972291 
.972245 

.78 

.77 
78 
.78 

77 

7R 

9.565373 
565763 
.566153 
566542 
.566932 
.567320 

6.50 
6.50 
6.48 
6.50 
6.47 
6  48 

10.434627 
.434237 
.433847 
.433458 
.433068 
.432680 

49 
48 
47 
46 
45 
44 

17 
18 

.  539907 
.540249 

5.70 
5.68 

.972198 
.972151 

.  in 

.78 

MM 

567709 
.568098 

6^48 

fi  47 

.432291 
.431902 

43 

42 

19 

.540590 

5.68 

.972105 

.  i  1 
r-o 

.568486 

O.4< 

.431514 

41 

20 

.540931 

5.68 
5.68 

.972058 

.  <O 

.78 

.568873 

6^47 

.431127 

40 

21 

9.541272 

9.972011 

fO 

9.569261 

6.45 

10.430739 

39 

22 

.541613 

5.68 

.971964 

.  to 

r<o 

.569648 

6AK 

.430352 

38 

23 

.541953 

5.67 

971917 

.  to 
7ft 

.570035 

.40 

6AK 

.429965 

37 

24 

.542293 

5.67 

971870 

.  to 
'"ft 

.570422 

.40 

6AK 

.429578 

36 

25 
26 

.  542632 
.542971 

5.65 
5.65 

.971823 
.971776 

.  to 

.78 

7ft 

.570809 
.571195 

.40 

6.43 

.429191 

.428805 

35 
34 

27 

.543310 

5.65 

.971729 

.  to 

.571581 

6A<\ 

.428419 

33 

2H 
29 
30 

.543649 
.543987 
.544325 

5.65 
5.63 
5.63 
5.63 

.971682 
.971635 
.971588 

.78 
.78 
.78 
.80 

.571967 
.572352 
.572738 

.40 

6.42 
6.43 
6.42 

.428033 
.427648 
.427'262 

32 
31 
30 

31 

9.544663 

9.971540 

7ft 

9.573123 

6  40 

10.426877 

29 

32 
33 

.545000 
.545338 

5.62 
5.63 

.971493 
.971446 

.  t  O 

.78 

.573507 
.573892 

6^42 

.426493 
,426108 

28 
27 

34 

35 

.545674 
.546011 

5.60 
5.62 

.971398 
.971351 

'.78 

Of) 

.574276 
.574660 

6^40 
6  40 

.425724 
.425340 

26 
25 

36 

.546347 

5.60 

.971303 

.oU 
7ft 

.575044 

6'  00 

.424956 

24 

37 

.546683 

5.60 

.971256 

.  to 

.575427 

.OO 
6OQ 

.424573 

23 

38 
39 

.547019 
.547354 

5.60 
5.58 

.971208 
.971161 

.80 
.78 
ft/i 

.575810 
1  .576193 

.  OO 

6.38 
6  38 

.424190  i  22 
.423807  21 

40 

.547689 

5  .  58 
5.58 

.971113 

.  cAJ 

.78 

1  .576576 

.423424 

20 

41 

9.548024 

9.971066 

9.576959 

607 

10.423041 

19 

42 

.548359 

5.58 

.971018 

.80 

.577341 

.01 

6  37 

.422659 

18 

43 

.548693 

5  57 

.970970 

.80 

PA 

.577723 

6  35 

.422277   17 

44 

.549027 

5.57 

.970922 

.oU 

QA 

.578104 

6  37 

.421896   16 

45 

46 

.549360 
.549693 

5.55 
5.55 

.970874 
.970827 

.oU 

.78 

.578486 
.578867 

6."35 

.421514 
.421133 

15 
14 

47 

48 

.550026 
.550359 

5.55 
5.55 

.970779 
.970731 

.80 
.80 

.579248 
579629 

6  .35 
6.35 

(•  OO 

.420752 
.420371 

13 
12 

49 
50 

.550692 
.551024 

5.55 
5.53 
5.53 

.970683 
970635 

80 

.80 
.82 

.580009 
.  580389 

D.oo 

6.33 
6.33 

.419991 
.419611 

11 
10 

51 

9.551356 

9.970586 

QA 

9.580769 

600 

10.419231 

9 

52 

.551687 

5.52 

.970538 

.oU 

PA 

.581149 

.00 

6  3'-* 

.418851 

8 

53 

.552C18 

5  .  52 

970490 

.  oO 

581528 

.418472 

7 

54 
55 

.552349 
.  552680 

5.52 
5.52 

.970442 
.970394 

.80 
.80 

ftO 

581907 
.582286 

6^32 
6  32 

.418093 
.417714 

6 
5 

56 

.553010 

5.50 

.970345 

.04 

.582665 

69,0 

.417335 

4 

57 

.553341 

5.52 

.97'0297 

.80 

.583044 

.o« 

6OA 

.416956 

3 

58 
59 

.553670 
.554000 

5.48 
5.50 

.970249 
.970200 

.80 

.82 

.583422 
.583800 

.oU 

6.30 

.416578 
.416200 

2 
1 

60 

9.554329 

5.48 

9.970152 

.80 

9.584177 

6.28 

10.415823 

0 

' 

Cosine. 

D.  1".  ! 

Sine.  | 

D.  r.  1 

Cotang. 

D.  1". 

Tang. 

/  i 

124 


69* 


21* 


TABLE   X. — LOGATUTHMIC   SINES, 


' 

Sine. 

D.  r. 

Cosine. 

D.  1'. 

Tang. 

D.  r. 

Cotang. 

' 

0 

9.554329   „  J0 

9.970152 

9.584177 

10.415823 

60 

1 

2 

.554658 
.554987 

O.<*0 

5.48 

.970103 

.  970055 

'.80 

.584555 
.584932 

6.30 
6.28 

6.)0 

.415445 
.415068 

59 
58 

3 
4 

.555315 
.555643 

5^47 

547 

.970006 
.969957 

'.82  ] 

.585309 
.585686 

./CO 

6.28 

6O7 

.414691 
.414314 

57 
56 

5 

.555971 

.It 

547 

.969909 

'DC! 

.586062 

>V< 

.413938 

55 

6 

.556299 

.4< 

5*K 

.969860 

.8,4    ; 

.586439 

6.28 

.413561 

54 

7 
8 
9 

.556626 
.556953 
.557280 

.40 

5.45 
5.45 

5  An 

.969811 
.969762 
.969714 

.82 
.82  i 
.80 

QO 

.586815 
.587190 
.587566 

6.27 
6.25 
6.27 

.413185 
.412810 
.412434 

53 
52 
51 

10 

.557606 

.4o 
5.43 

.909665 

.82 
.82  | 

.587941 

6.25 
6.25 

.412059 

50 

11 

12 

9.557932 
.558258 

5.43 

9.969616 

.9695(57 

.82 

9.588316 
.588691 

6.25 

10.411684 
.411309 

49 
48 

13 

.558583 

5.42 

5451 

.969518 

.  .82 

GO 

.589066 

6.25 

6QO 

.410934 

47 

14 

.558909 

.4o 

.969469 

.04 

.589440 

.29 

.410560 

46 

15 

.559234 

5.42 

.969420 

.82  i 

.589814 

6.23 

.410186  !  45 

16 
17 

18 

.559558 
.559883 
.560207 

5.40 
5.42 
5.40 

.969370 
.969321 
.969272 

.83  : 
.82 
.82  ! 

.590188 
.590562 
.590935 

6.23 
6.23 
6.22 

.409812 
.409438 
.409065 

44 
43 
42 

19 

.560531 

5.40 

.969223 

.82 

.591308 

6.22 

.408692 

41 

20 

.560855 

5.40 
5.38 

.969173 

.83 
.82 

.591681 

6.22 
6.22 

.408319 

40 

21 

9.561178 

9.969124 

9.592054 

10.407946 

39 

22 
23 

.561501 
.561824 

5.o8 
5.38 

.969075 
.969025 

.82 

.83  ! 

.592426 
.592799 

6.20 
6.22 

.407574 
.407201 

38 
37 

24 
25 
26 

.562146 
.562468 
.  562790 

5.37 
5.37 
5.37 

.968976 
.968926 
.968877 

!as 

.82  : 

.593171 
.593542 
.593914 

6^18 
6.20 
6ifl 

.406829 
.406458 
.406086 

36 
35 
34 

27 
28 
29 

.563112 
.563433 
.563755 

5.37 
5.35 
5.37 

.968827 
.968777 
.968728 

.83 
.83 
.82  ! 

.594285 
.594656 
.595027 

.  lo 

6.18 
6.18 

.405715 
.405344 
.404973 

33 
32 
31 

30 

.564075 

5.33 
5.35 

.968678 

.83 
.83 

.595398 

6.18 
6.17 

.404602 

30 

31 

9.564396 

9.968628 

9.595768 

10.404232 

29 

32 

.564716 

5.33 

.968578 

.83 

.596138 

6.17 

617 

.403862 

28 

33 

.565036 

5.33 

.968528 

.83  . 

.596508 

.If 

.403492 

27 

34 

.565356 

5.33 

.968479 

.82 

.596878 

6.17 

.403122 

26 

35 

.565676 

5.33 

.968429 

.83 

.5G7247 

6.15 

61K 

.402753 

25 

36 

.565995 

5.32 

.968379 

.83  ! 

.597616 

.15 

.402384 

24 

37 

.566314 

5.32 

.968329 

.83 

.597985 

6.15 

.402015 

23 

38 

.566632 

5.30 

.968278 

.85 

.59&S54 

6.15 

.401646 

22 

39 

.566951 

5.32 

.968228 

.83    ; 

.598722 

6.13 

.401278 

21 

40 

.567269 

5.30 
5.30 

.968178 

.83  i 
.83 

.599C91 

6.15 
6.13 

.400909 

20 

41 
42 

9.567587 
.567904 

5.28 

9.968128 
.968078 

.83 

9.599459 
.599827 

6.13 

10.400541 
.400173 

19 

18 

43 

.568222 

.968027 

.85  ; 

.600194 

6.12 

.399806 

17 

44 
45 

.568539 
.568856 

•        Qfir'Q77 
K  OQ      .JQiJii 

£07     .967927 

.83 
.83  ; 

.600562 
.600929 

e!i2 

.399438 
.399071 

16 
15 

46 

.569172 

g-^    .967876 

.85  i 

OQ    i 

.601296 

6.12 
6  12 

.398704 

14 

47 
48 
49 
50 

.569488 
.569804 
.570120 
.570435 

5^27 
5.27  i 
5.25 
5.27 

.967826 
.967775 
.967725 
.967674 

.OO 

.85 
.83  | 

.85-  ! 
.83  ! 

.601663 
.602029 
.602395 
.602761 

eiio 

6.10 
6.10 
6.10 

.398337 
.397971 
.397605 
.397239 

13 
12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.570751 
.571066 
.571380 
.571695 
.572009 
.572323 
.572636 
.572950 
.573263 
9.573575 

5.25 
5.23 
5.25 
5.23 
5.23 
5.22 
5.23 
5.22 
5.20 

9.967624 
.967573 
.967522 
.967471 
.967421 
.967370 
.967319 
.967268 
.967217 
i  9.967166 

.85 
.85  1 
.85 
.83 
.85 
.85 
.85 
.85 
.85 

9.603127 
.603493 
.603858 
.604223 
.604588 
.604953 
.605317 
.605682 
.606046 
9.606410 

6.10 

6.08 
6.08 
6.08 
6.08 
6.07 
6.08 
6.07 
6.07 

10.396873 
.396507 
.396142 
.395777 
.395412 
.395047 
.394683 
.394318 
.393954 
10.393590 

9 

8 

6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  1".    Sine,  i  D.  1'. 

Cotang. 

D.  r. 

Tang.  1  ' 

111' 


125 


COSINES,  TANGENTS,  AND  COTANGENTS. 


' 

Sine. 

D.  1". 

Cosine. 

D.  r.  1 

Tang. 

D.  r. 

Cotang. 

' 

0 

9.573575 

599 

9.967166 

QK 

9.606410 

ft  fiK 

10.393590 

60 

1 

2 
3 

.573888 
.574200 
.574512 

.as 

5.20 
5.20 
5  20 

.967115 
..967064 
.967013 

.OO 

.85 
.85. 

87 

.  606773 
.607137 
.607500 

O  .UO 

6.07 
6.05 

6nc 

.393227 
.392863 
.392500 

59 
58 
57 

4 
5 
6 

.574824 
.575136 
.575447 

5^20 
5.18 

518 

.966961 
.966910 
.966859 

.o< 

.85 

.85 

OK 

.607863 
.608225 
.608588 

.UO 

6.03 
6.05 

6  no 

.392137 
.391775 
.391412 

56 
55 
54 

7 

.575758 

.  lo 

.966808 

.00 

.608950 

.Uo 

.391050 

53 

8 

.576069 

5.18 

517 

.966756 

.87 

OK 

.609312 

6.03 
6  no 

.390688 

52 

9 
10 

.576379 
.576689 

.  If 

5.17 
5.17 

.966705 
.966653 

.00 

.87 
.85 

.609674 
.610036 

.Uo 

6.03 
6.02 

.390326 
.389964 

51 
50 

11 

12 

9.576999 
.577309 

5.17 

9.966602 
.966550 

.87 

9.610397 
.610759 

6.03 

10  389603 
.389241 

49 

48 

13 
14 
15 

.577618 
577927 
.578236 

5.15 
5.15 
5  15 

.966499 
.966447 
.966395 

.85 

.87 
.87 

.611120 
.611480 
.611841 

6.02 
6.00 
6.02 

.388880 
.388520 
.388159 

47 
46 
45 

16 

17 
18 

.578545 

.578853 
.579162 

5.15 
5.13 
5.15 
51°. 

.966344 
.966292 
.966240 

.85 
.87 
.87 

87 

.612201 
.612561 
.612921 

6.00 
6.00 

6.00 
61  r\ 

.387799 
.387439 
.387079 

44 
43 

42 

19 

.579470 

.  10 

.966188 

.  01 

.613281 

.1  U 

.386719 

41 

20 

.579777 

5.12 
5.13 

.966136 

.87 
.85 

.613641 

6.00 
5.98 

.386359 

40 

21 

9.580085 

519 

9.966085 

87 

9.614000 

10.386000 

39 

22 
23 
24 

.580392 
.580699 
.581005 

.  I  -v 

5.12 
5.10 

.966033 
.965981 
.965929 

.Of 

.87 
.87 

QQ 

.614359 
.614718 
.615077 

5^98 
5  98 

K  G7 

.385641 
.385282 
384923 

38 
37 
36 

25 
26 
27 
28 
29 

.581312 
.581618 
.581924 
.582229 
.582535 

5.12 
5.10 
5.10 
5.08 
5.10 

.965876 
.965824 
.965772 
.965720 
.965668 

.OO 

.87 
87 
.87 
.87 

.615435 
.615793 
.616151 
.616509 
616867 

5^97 
5.97 
5.97 
5.97 

.384565 
.384207 
.383849 
.383491 
.383133 

35 
34 
33 
32 
31 

30 

.582840 

5.08 
5.08 

.965615 

.88 
.87 

.617224 

5  95 
5  97 

.382776 

30 

31 
32 

9.583145 
.583449 

5.07 

5  no 

9  965563 
965511 

.87 

00 

9.617582 
617939 

5.95 

K  no 

10  382418 
.382061 

20 

28 

33 
34 

583754 
.584058 

.Uo 
5.07 

5npr 

965458 
965406 

.00 

.87 

88 

.618295 
.  618652 

o.yo 
5.95 

K  QQ 

.381705 
.381348 

27 
26 

35 
36 

.584361 
.584665 

.UO 

5.07 

5ne 

965353 
965301 

.00 

.87 

88 

619008 
.619364 

o.yo 
5.93 

5QO 

.380992 
.380636 

25 

24 

37 

38 
39 

.584968 
.585272 
.585574 

.UO 

5.07 
5.03 

965248 
.965195 
.965143 

.00 

88 
.87 

.619720 
.620076 
.620432 

.yo 
5.93 
5  93 

.380280 
.379924 
.379568 

23 

22 
21 

40 

.585877 

5.05 
5.03 

.965090 

.88 
.88 

.620787 

5.92 
5.92 

.379213 

20 

41 

9.586179 

5/Vr 

9.965037 

QQ 

9.621142 

10.378858 

19 

42 
43 
44 
45 

.586482 
.586783 
.587085 
.587386 

.UO 

5.02 
5.03 
5.02 

5  no 

.964984 
.964931 
.964879 
.964826 

.OO 

.88 
.87 
88 

88 

.621497 
.621852 
.622207 
.622561 

5^92 
5.92 
5.90 
5  on 

.378503 
.378148 
.377793 
.377439 

18 
17 
16 
15 

46 

.587688 

.Uo 

964773 

.00 

.622915 

.yu 

.377085 

14 

47 

48 

.587989 
.588289 

5.02 
5.00 

.9647'20 
964666 

.88 
.90 

.623269 
.623623 

5.90 
5.90 

.376731 
.376377 

13 

12 

49 
50 

.588590 
.588890 

5.02 
5.00 
5.00 

964613 
.964560 

.88 
-.88 
.88 

.623976 
.624330 

5.88 
5.90 
5.88 

.376024 
.375670 

11 
10 

51 

9.589190 

9.964507 

88 

9.624683 

t  QQ 

10.375317 

9 

52 
53 

.589489 
.589789 

5^00 

.964454 
.964400 

.00 

.90 

.625036 

.625388 

0  .00 

5.87 

.374964 
.374612 

8 

54 

.590088 

4.98 

.964347 

.88 

.625741 

5.88 

.374259 

6 

55 

56 

.590387 
.590686 

4.98 
4.98 

.964294 
.964240 

.88 
.90 

.626093 
.626445 

5.87 
5.87 

.373907 
.373555 

5 

4 

57 

58 

590984 
591282 

4.97 
4.97 

.964187 
.964133 

.88 
.90 

.626797 
.627149 

5.87 
5.87 

.373203 
.372851 

3 

2 

59 
60 

.591580 
9.591878 

4.97 
4.97 

964080 
9.964026 

.88 
.90 

.627501 
9.627852 

5.87 
5.85 

.372499 
10.372148 

1 
0 

1 

Cosine. 

D.  1". 

Sine. 

D.  1".  | 

Cotang. 

D.  1". 

Tang. 

' 

126 


67° 


TABLE    X. — LOGARITHMIC    SINES, 


Sine. 

D.  r. 

Cosine. 

D.  1*. 

Tang. 

D.  r. 

Cotang. 

' 

0 

1 

9.591878 
.592176 

4.97 

4QK 

9.964026 
.963972 

.90 

QQ 

9.627852 
.628203 

5.85 

5  OK 

10.372148 
.371797 

60 
59 

2 

3 
4 
5 

.592473 
.592770 
.593067 
.593363 

.yo 
4.95 
4.95 
4.93 

.963919 
.963865 
.963811 
.963757 

.OO 

.90 
.90 
.90 

.62&5S4 
.628905 
.629255 
.629606 

.OD 

5.85 

5.83 
5.85 

.371446 
.371095 
.370745 
.370394 

58 
57 
56 
55 

6 

.593659 

4.93 

4  nO 

.963704 

.88 

QO 

.629956 

5.83 

5  go 

.370044 

54 

7 
8 
9 

.593955 
.594251 
.594547 

.yo 
4.93 
4.93 

.963650 
.963596 
.963542 

.yu 
.90 
.90 

QO 

.630306 
.630656 
.631005 

.OO 

5.83 

5.82 

500 

.369694 
.369344 
.368995 

53 

52 

51 

10 

.594842 

4i92 

.963488 

.yu 
.90 

.631355 

.00 

5.82 

.368645 

50 

11 
12 

9.595137 
.595432 

4.92 
4  92 

9.963434 
.963379 

.92 

QO 

9.631704 
.632053 

5.82 

5QO 

10.368296 
.367947 

49 

48 

13 

.595727 

4QO 

.963325 

.yu 

QO 

.632402 

.O/* 
5P.O 

.367598 

47 

14 

.596021 

.yu 

4Q.fl 

.963271 

.yu 

Qfl 

.632750 

.oU 

5QO 

.367250 

46 

15 
16 

.596315 
.596609 

.yu 
4.90 
4  90 

.963217 
.963163 

.yu 
.90 
92 

.633099 
.633447 

.o4 
5.80 
5  80 

.366901 
.366553 

45 
44 

17 
18 
19 

.596903 
.597196 
.597490 

4^88 
4.90 

4QQ 

.963108 
.963054 
.962999 

'.90 
.92 
90 

.633795 
.634143 
.634490 

s!so 

5.78 
5  80 

.366205 
.365857 
.365510 

43 
42 
41 

20 

.597783 

.OO 

4.87 

.962945 

'.92 

.634838 

5l  78 

.365163 

40 

21 
22 
23 
24 
25 
26 
27 
28 

9.598075 
.598368 
.598660 
.598952 
.599244 
.599536 
.599827 
.600118 

4.88 
4.87 
4.87 
4.87 
4.87 
4.85 
4.85 

9.962890 
.962836 
.962781 
.962727 
.962672 
.962617 
.962562 
.962508 

.90 
.92 
.90 
.92 
.92 
.92 
.90 

9.635185 
.635532 
.635879 
.636226 
.636572 
.636919 
.637265 
.637611 

5.78 
5.78 
5.78 
5.77 
5.78 
5.77 
5.77 

10.364815 
.364468 
.364121 
.363774 
.363428 
.363081 
.362735 
.362389 

39 
38 
37 
36 
35 
34 
33 
32 

29 

.600409 

4.85 
4QK. 

.962453 

.92 

.637956 

5.75 

577 

.362044 

31 

30 

.600700 

.00 

.962398 

,\)4i 

.638302 

.  i  i 

.361698 

30 

4.83 

.92 

5.75 

31 

9.600990 

4QO 

9.962343 

9.638647 

Si^K 

10.361353 

29 

32 
33 

.601280 
.601570 

.OO 

4.83 

4QQ 

.962288 
.962233 

!92 
no 

.638992 
.639337 

.  <o 
5.75 

57K 

.361008 
.360663 

28 
27 

34 

.601860 

.OO 

.962178 

.639682 

.  to 

.360318 

26 

35 

.602150 

4.83 

.962123 

.92 

QO. 

.640027 

5.75 
5  73 

.359973 

25 

36 
37 

.602439 
.602728 

4^82 

4QO 

.962067 
.962012 

.yo 
.92 

.640371 
.640716 

5l  75 
5  73 

'.  359284 

24 
23 

38 
39 
40 

.603017 
.603305 
.603594 

.O* 

4.80 
4.82 
4.80 

.961957 
.961902 
.961846 

192 
.93 
.92 

.641060 
.641404 
.641747 

5.  '73 
5.72 
5.73 

.358940 
.358596 
.358253 

22 
21 
20 

41 

9.603882 

4fto 

9.961791 

9.642091 

5  72 

10.357909 

19 

42 

.604170 

.ou 

.961735 

•J5 

.642434 

.357566 

18 

43 

.604457 

4.78 

.961680 

.92 

.642777 

5.72 

5rv> 

.357223 

17 

44 

45 

.604745 
.605032 

4  '.78 

.961624 
.961569 

!92 

.643120 
.643463 

.  (  -w 

5.72 

.356880 
.356537 

16 
15 

46 

47 
48 

.605319 
.605606 
.605892 

4.78 
4.78 

4.77 

.961513 
.961458 
.961402 

.93 
.92 
.93 

QO 

.643806 
.644148 
.644490 

5.72 
5.70 
5.70 

57O 

.356194 
.355852 
.355510 

14 
13 

12 

49 

.606179 

A*? 

.961346 

.yo 

QO. 

.644832 

.  i  U 
5r-/"» 

.355168 

11 

50 

.606465 

4.77 

.961290 

.yo 

.92 

.645174 

.  <u 
5.70 

.354826 

10 

51 

9.606751 

4**K 

9.961235 

9.645516 

5po 

10.354184 

9 

52 
53 

.607038 
.607322 

.  t  •  > 

4.77 

.961179 
.961123 

'.98 

.645857 
.646199 

.  Do 

5.70 

.354143 

.353801 

8 

54 

.607607 

4.75 

.961067 

.93 

QO 

.646540 

5.68 

.353460 

6 

55 

56 

.607892 
.608177 

4.75 
4.75 

47°. 

.961011 
.960955 

.  yo 
.93 

.646881 
.647222 

5.  '68 

5fi7 

.353119 

.352778 

5 
4 

57 

.608461 

.  to 

.960899 

'OQ 

.647562 

.  Oi 

Sfift 

.352438 

3 

58 
59 
60 

.608745 
.609029 
9.609313 

4.73 
4.73 
4.73 

.960843 
.960786 
9.960730 

.yo  ; 
.95 
.93 

.647903 
.648243 
9.648583 

.  oo 
5.67 
5.67 

.352097 
.351757 
10.351417 

2 
1 
0 

' 

I  Cosine. 

D.I*. 

Sine. 

D.  1".  j 

|  Cotang. 

D.  r. 

Tang. 

' 

127 


COSINES,  TANGENTS,  AND  COTANGENTS. 


155* 


' 

Sine. 

D.  1". 

Cosine. 

D.  r. 

Tang. 

D.1-. 

Cotang. 

' 

0 

1 

9.609313 
.609597 

4-£3 

9.960730 
.960674 

.93 

9.648583 
.648923 

5.67 

10.351417 
.351077 

60 
59 

2 

.609880 

Ho     .960618 

.93 

QK 

.649263 

5.67 

5fiK 

.350737 

58 

3 

.610164 

\'l%    .960561 

.yo 

QO 

.649602 

.DO 

5  cry 

.350398 

57 

4 

.610447 

J'2!   1  .960505 

.yo 

QK 

.649942 

.Of 

5f»K 

.350058 

56 

5 

.610729 

4.  <U 

.960448 

.yo 

.650281 

.DO 

.349719 

55 

6 

.611012 

4.72 

.960392 

.93 

.650620 

5.65 

.349380 

54 

7 

.611294 

4.70 

.960335 

.95 

QO. 

.650959 

5.65 

Son 

.349041 

53 

8 

.611576 

^  iJJ 

.960279 

.yo 

.651297 

.DO 

.348703 

52 

9 
10 

.611858 
.612140 

4.70 
4.70 
4.68 

.960222 
.960165 

.95 
.95 
.93 

.651636 
.651974 

5.65 
5.63 
5.63 

.348364 
.348026 

51 
50 

11 

9.612421 

9.960109 

9.652312 

10.347688 

49 

12 

.612702 

4.  bo 

.960052 

QK 

.652650 

5  ftp. 

.347350 

48 

13 
14 

.612983 
.613264 

4^68 

.959995 
.959938 

.yo 
.95 

QO 

.652988 
.653326 

.DO 

5.63 

.347012 
.346674 

47 
46 

15 
16 
17 
18 
19 
20 

.613545 
.613825 
.614105 
.614385 
.614665 
.614944 

4.68 
4.67 
4.67 
4.67 
4.67 
4.65 
4.65 

.959882 
.959825 
.959768 
.959711 
.959654 
.959596 

.yd 
.95 
.95 
.95 
.95 
.97 
.95 

.653663 
.654000 
.654337 
.654674 
.655011 
.655348 

5.62 
5.62 
5.62 
5.62 
5.62 
5.62 
5.60 

.346337 
.346000 
.345663 
.345326 
.344989 
.344652 

45 
44 
43 
42 
41 
40 

21 
22 

23 
24 
25 
26 

27 
28 
29 

9.615223 
.615502 
.615781 
.616060 
.616338 
.616616 
.616894 
.617172 
.617450 

4.65 
4.65 
4.65 
4.63 
4.63 
4.63 
4.63 
4.63 

9.959539 
.959482 
.959425 
.959368 
.959310 
.959253 
.959195 
.959138 
.959080 

.95 
.95 
.95 
.97 
.95 
.97 
.95 
.97 
95 

9.655684 
.656020 
.656356 
.656692 
.657028 
.657364 
.657699 
.658034 
.658369 

5.60 
5.60 
5.60 
5.60 
5.60 
5.58 
5.58 
5.58 

5  CO 

10.344316 
.343980 
.343644 
.343308 
.342972 
.342636 
.342301 
.341966 
.341631 

39 
38 
37 
36 
35 
34 
33 
32 
31 

30 

.617727 

4^62 

.959023 

.658704 

.Oo 

5.58 

.341296 

30 

31 

9.618004 

4  ing 

9.958965 

QK 

9.659039 

5K7 

10.340961 

29 

32 
33 
34 

.618^8! 
.618558 
.618834 

,u« 

4.62 
4.60 

4  Aft 

.958908 
.958850 
.958792 

.yo 

.97 
.97 

.659373 
.659708 
.660042 

.Of 

5.58 
5.57 

.340627 
.340292 
.339958 

28 
27 
26 

35 
36 

.619110 
.619386 

.  Ov 

4.60 

.958734 
.958677 

'.95 

.660376 
.660710 

5.57 
5.57 

.339624 
.&39290 

25 
24 

37 
38 
39 
40 

.619662 
.619938 
.620213 
.620488 

4.60 
4.60 
4.58 
4.58 
4.58 

.958619 
.958561 
.958503 
.958445 

.97 
.97 
.97 
.97 
.97 

.661043 
.661377 
.661710 
.662043 

5.55 
5.57 
5.55 
5.55 
5.55 

.338957 
.338623 
.338290 
.337957 

23 
22 
21 
20 

41 
42 
43 
44 

9.620763 
.621038 
.621313 

.621587 

4.58 
4.58 
4.57 

4K7 

9.958387 
.958329 
.958271 
.958213 

.97 
.97 
.97 

9.662376 
.662709 
.663042 
.663375 

5.55 
5.55 
5.55 

5KQ 

10.337624 
.337291 
.336958 
.386625 

19 
18 
17 
16 

45 
46 

47 

.621861 
.622135 
.622409 

.Of 

4.57 
4.57 

.958154 
.958096 
.958038 

!97 
.97 

.663707 
.664039 
.664371 

.00 
5.53 
5.53 

.336293 
.335961 
.335629 

15 
14 
13 

48 
49 

50 

.622682 
.622956 
.623229 

4.55 
4.57 
4.55 
4.55 

.957979 
.957921 
.957863 

.98 
.97 
.97 
.98 

.664703 
.6650.35 
.665366 

5.53 
5.53 
5.52 
5.53 

.335297 
.334965 
.334634 

12 
11 
10 

51 
52 

9.62,3502 
.623774 

4.53 

9.957804 
.957746 

.97 

9.665698 
.666029 

5.52 

10.334302 
.333971 

9 
8 

53 

.624047 

4.55 

.957687 

.98 

.666360 

5.52 

.333640 

7 

54 
55 
56 
57 
58 
59 
60 

.624319 
.624591 
.624863 
.625135 
.625406 
.625677 
9.625948 

4.53 
4.53 
4.53 
4.53 
4.52 
4.52 
4.52 

.957628 
.957570 
.957.11 
.957452 
.957393 
.957335 
9.957'276 

.98 
.97 
.98 
.98 
.98 
.97 
.98 

.666691 
.667021 
.667352 
.667682 
.668013 
.668343 
9.668673 

5.52 
5.50 
5.52 
5.50 
5.52 
5.50 
5.50 

.333309 
.332979 
.332648 
.332318 
.331987 
.331657 
10.331327 

6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  1".  ;   Sine. 

D.  1". 

Cotang. 

D.  1". 

Tang. 

1 

128 


25° 


TABLE   X. — LOGARITHMIC   SIKES, 


154" 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

9.625948 
.626219 

4.52 

9.957276 
.9572^7 

.98 

9.G68673 
.669002 

5.48 

10.331327 
.330998 

60 
59 

2 

.626490 

4.52 

.957158 

.98 

.669332 

5.50 

.330668 

58 

3 

4 
5 
6 

8 

.626760 
.627030 
.627300 
.627570 
.627840 
.628109 

4.50 
4.50 
4.50 
4.50 
4.50 
4.48 

.957099 
.957040 
.956981 
.956921 
.956862 
.956803 

.98 
.98 
.98 
1.00 
.98 
.98 

QQ 

.669661 
.669991 
.670320 
.670649 
.670977 
.671306 

5.48 
5.50 
5.48 
5.48 
5.47 
5.48 

.330339 
.330009 
.329680 
.329351 
.329023 
.328694 

57 
56 
55 
54 
53 
52 

9 
10 

.628378 
.628647 

4.  '48 
4.48 

.956744 
.956684 

.yo 
1.00 
.98 

.671635 
.671963 

5.48 
5.47 
5.47 

.328365 
.328037 

51 

50 

11 

9.628916 

9.956625 

9.672291 

10.327709 

49 

13 

13 
14 
15 
16 
17 
18 
19 
20 

.629185 
.629453 
.629721 
.629989 
.630257 
.630524 
.630792 
.631059 
.631326 

4^47 
4.47 
4.47 
4.47 
4.45 
4.47 
4.45 
4.45 
4.45 

.956566 
.956506 
.956447 
.956387 
.956327 
.956268 
.956208 
.956148 
.956089 

i!oo 

.98 
1.00 
1.00 
.98 
1.00 
1.00 
.98 
1.00 

.672619 
.672947 
.673274 
.673602 
.673929 
.674257 
.674584 
.674911 
.675237 

5.47 
5  47 
5.45 
5.47 
5.45 
5.47 
5.45 
5.45 
5.43 
5.45 

.327381 
.327053 
.326726 
.326398 
.326071 
.325743 
.325416 
.325089 
.324763 

48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.631593 
.631859 
.632125 
.632392 
.632658 
.632923 
.633189 
.633454 
.633719 
.633984 

4.43 
4.43 
4.45 
4.43 
4.42 
4.43 
4.42 
4.42 
4.42 
4.42 

9.956029 
955969 
.955909 
.955849 
.955789 
.955729 
.955669 
.955609 
.955548 
.955488 

1.00 
1.00 
1.00 
1.00 
1.00 
1.00 
1.00 
.98 
1  00 
1.00 

9.675564 
.675890 
.676217 
.676543 
.676869 
.677194 
.677520 
.677846 
.678171 
.678496 

5  43 
5.45 
5.43 
5.43 
5.42 
5.43 
5.43 
5.42 
5  .42 
5.42 

10.324436 
.324110 
.323783 
.323457 
.323131 
.322806 
.322480 
.322154 
.321829 
.321504 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 

32 
33 
34 
35 
36 
37 
38 
39 
40 

9.634249 
.634514 
.634778 
635042 
.635306 
635570 
.635834 
.636097 
636360 
.636623 

4.42 
4.40 
4.40 
4.40 
4.40 
4.40 
4.38 
438 
4.38 
4.38 

9  955428 
.955368 
.955307 
.955247 
.955186 
.955126 
.955065 
.955005 
.954944 
.954883 

1.00 
1.02 
1.00 
1.02 
1.00 
1.02 
1.00 
1.02 
1.02 
1.00 

9.678821 
679146 
.679471 
.679795 
.680120 
.680444 
.680768 
.681092 
.681416 
.681740 

5.42 
5  42 
5.40 
5.42 
5.40 
5.40 
5.40 
5.40 
5.40 
5.38 

10.321179 
.320854 
.320529 
.320205 
.319880 
.319556 
.319232 
.318908 
.318584 
.318260 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 

9.636886 
637148 
.637411 
.637673 

4.37 
4.38 
4.37 

9  954823 
.954762 
.954701 
.954640 

1  02 
1.02 
1.02 

9.682063 
.682387 
.682710 
.683033 

5.40 
5.38 
5.38 

5  no 

10  317937 
.317613 
.317290 
.316967 

19 
18 
17 
16 

45 

46 
47 
48 
49 
50 

.637935 
.638197 
.638458 
.638720 
.638981 
.639242 

4.37 
4.37 
4.35 
4.37 
4.35 
4.&5 
4.35 

.954579 
.954518 
.954457 
.954396 
.954335 
.954274 

1.02 
1.02 
1.02 
1  02 
1  02 
1.02 
1.02 

.683356 
.683679 
.684001 
.684324 
.684646 
.684908 

OO 

5.38 
5.37 
5.38 
5.37 
5.37 
5.37 

.316644 
.316321 
.315999 
.315676 
.315354 
.315032 

15 
14 
13 
12 
11 
10 

51 

52 

9.639503 
.639764 

4.35 

9954213 
.954152 

1.02 

9.685290 
.685612 

5.37 

507 

10  314710 
314388 

9 

8 

53 
54 
55 
56 
57 
58 
59 

.640024 
.640284 
.640544 
.640804 
.641064 
.641324 
.641583 

4.33 
4.33 
4.33 
4.33 
4.33 
4.33 
4.32 

.954090 
.954029 
.953968 
.953906 
.953845 
.953783 
.953722 

1  .03 
1.02 
1.02 
1  03 
1.02 
1.03 
1.02 

.C85934 
R86255 
.686577 
.686898 
.687219 
.687540 
.687861 

.0< 

5.35 
5.37 
5  35 
5.35 
5  35 
5.35 

314066 
313745 
.313423 
.313102 
.312781 
.312460 
.312139 

7 
6 
5 
4 
3 
2 
1 

60 

9.641842 

4.32 

9.953660 

1.03 

9.688182 

S'35   10.311818 

0 

' 

Cosine. 

D.  1'. 

Sine.  1  D.  1'.  1 

Cotang. 

D.  1".    Tang. 

1 

64* 


19.Q 


26' 


COSINES,   TANGENTS,   AND  COTANGENTS. 


' 

Sine. 

D.  1'. 

Cosine. 

D.  1*. 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 
3 
4 
5 
6 

9.641842 
.642101 
.642360 
.642618 
.642877 
.643135 
.643393 

4.32 
4.32 
4.30 
4.32 
4.30 
4.30 

9.953660 
.953599 
.953537 
.953475 
.953413 
.953352 
.953290 

1.02 
1.03 
1.03 
1.03 
1.02 
1.03 
1  03 

9.688182 
.688502 
.688823 
.689143 
.689463 
.689783 
.690103 

5.33 
5.32 
5.33 
5.33 
5.33 
5.33 

5OQ 

10.311818 
.311498 
.311177 
.310857 
.310537 
.310217 
.309897 

60 
59 
58 
57 
56 
55 
54 

7 
8 
9 
10 

.643650 
.643908 
.644165 
.644423 

4.  '30 
4.28 
4.30 
4.28 

.953228 
.953166 
.953104 
.953042 

1.'03 
1.03 
1.03 
1.03 

.690423 
.690742 
.691062 
.691381 

.OO 

5.32 
5.33 
5.32 
5.32 

.309577 
.309258 
.308938 
,308619 

53 
52 
51 
50 

11 

9.644680 

497 

9.952980 

9.691700 

509 

10.308300 

49 

12 

.644936 

.at 

.952918 

1  05 

1  .692019 

.0,4 

,307981 

48 

13 
14 
15 
16 
17 
18 
19 
20 

.645193 
.645450 
.645706 
.645962 
.646218 
.646474 
.646729 
.646984 

4^28 
4.27 
4.27 
4.27 
4.27 
4.25 
4.25 
4.27 

.952855 
.952793 
.952731 
.952669 
.952606 
.952544 
.952481 
.952419 

1.'03 
1.03 
1.03 
1.05 
1.03 
1.05 
1.03 
1.05 

.6923^8 
.692656 
.692975 
.693293 
.693612 
.693930 
.694248 
.694566 

5i30 
5.32 
5.30 
5.32 
5.30 
5.30 
5.30 
5.28 

.307662 
.307344 
.307025 
.306707 
.306388 
.306070 
.305752 
.305434 

i  47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 

9.647240 
.647494 
.647749 
.648004 

4.23 
4.25 
4.25 
490 

9.952356 
.952294 
.952231 
.952168 

1.03 
1.05 
1.05 
1  03 

9.694883 
.695201 
.695518 
.695836 

5.30 

5  28 
5.30 

10.305117 
.304799 
.304482 
.304164 

39 

38 
37 
36 

25 

.648258 

.40 

A  9Q 

.952106 

.696153 

5.28 

59ft 

.303847 

35 

26 
27 
28 
29 
30 

.648512 
.648766 
.649020 
.649274 
.649527 

Q./Co 

4.23 
4.23 
4.23 
4.22 
4.23 

.952043 
.951980 
.951917 
.951854 
.951791 

l'05 
1.05 
1.05 
1.05 
1.05 

.696470 
.696787 
.697103 
.697420 
.697736 

./•CO 

5  28 
5.27 
5.28 
5.27 
5.28 

.303530 
.303213 
.302897 
.302580 
.302264 

34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.649781 
.650034 
.650287 
.650539 
.650792 
.651044 
.651297 
.651549 
.651800 
.652052 

4.22 
4.22 
4.20 
4.22 
4.20 
4.22 
4.20 
4.18 
4.20 
4.20 

9.951728 
.951665 
.951602 
.951539 
.951476 
.951412 
.951349 
.951286 
.951222 
.951159 

1.05 
1  05 
1.05 
1.05 
1.07 
1.05 
1  05 
1.'07 
1.05 
1.05 

9.698053 
.698369 
.698685 
.699001 
.699316 
.699632 
.699947 
.700263 
.700578 
.700893 

5.27 
5.27 
5.27 
5.25 
5.27 
5.25 
5.27 
5.25 
5.25 
5.25 

10.301947 
.301631 
.301315 
.300999 
.300684 
.300368 
.300053 
.299737 
.299422 
.299107 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 

9.652304 
.652555 
.652806 
.653057 
653308 
.653558 

4.18 
4.18 
4.18 
4.18 
4.17 

417 

9.951096 
.951032 
.950968 
.950905 
.950841 
.950778 

1.07 
1.07 
1.05 
1.07 
1.05 
1  07 

9.701208 
.701523 
.701837 
.702152 
.702466 
.702781 

5.25 
5.23 
5.25 
5.23 
5.25 

10.298792 

.298477 
.298163 
.297848  ' 
.297534  j 
.297219 

19 
18 
17 
16 
15 
14 

47 
48 
49 
50 

.653808 
.654059 
.654309 
.654558 

.  1  i 

4.18 
4.17 
4.15 

4.17 

.950714 
.950650 
.950586 
.950522 

l!07 
1.07 
1.07 
1.07 

.703095 
.703409 
.703722 
.704036 

5^23 
5.22 
5.23 
5.23 

.296905  • 
.296591  , 
.296278 
.295964 

13 

12 
11 
10 

51 
52 
53 
54 

55 
56 
57 

58 

9.654808 
.655058 
.655307 
.655556 
.655805 
.656054 
656302 
.656551 

4.17 
4.15 
4.15 
4.15 
4.15  i 
4.13 
4.15 

9.950458 
.950394 
.950330 
.950266 
.950202 
.950138 
.950074 
.950010 

1.07 
1.07 
1.07 
1.07 
1.07 
1.07 
1.07 

9.704350 
.704663 
.704976 
.705290 
.705603 
.705916 
.706228 
.706541 

5.22 
5.22 
5.23 
5.22 
5.22 
5.20 
5.22 

10.295650 
.295337 
.295024  ! 
.294710 
.294397 
.294084 
.293772 
.293459 

9 
8 
7 
6 
5 
4 
3 
2 

59 
60 

.656799 
9.657047 

4.13 
4.13 

.949945 
9.949881 

1.'07 

.706854 
9.707166 

5.  '20 

.293146 
10.292834 

1 
0 

' 

Cosine.  1  D.  1".  i 

Sine.   D.  1".  \  Cotang. 

D.  r. 

Tang. 

' 

TABLE    X. — LOGARITHMIC    SINES. 


152° 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 
3 

9.657047 
.657295 
.657542 
.657790 

4.13 
4.12 
4.13 

9.949881 
.949816 
.949752 

.949688 

1.08 
1.07 
1.07 

1  AS 

9.707166 
.707478 
.707790 
.708102 

5.20 
5.20 
5.20 

Son 

10.  292834 
.292522 

.  292210 
.291898 

60 
59 
58 
57 

4 

.658037 

.949623 

.708414 

5OA 

.291586 

56 

5 
6 

7 
8 
9 
10 

.658284 
.658531 
.658778 
.659025 
.659271 
.659517 

4.12 
4.12 
4.12 
4.10 
4.10 
4.10 

.949558 
.949494 
.949429 
.949364 
.949300 
.949235 

1.07 
1.08 
1.08 
1.07 
1.08 
1.08 

.708726 
.709037 
.709349 
.709660 
.709971 
.710282 

5.18 
5.20 
5.18 
5.18 
5.18 
5.18 

.291274 
.290963 
.290651 
.290340 
.290029 
.289718 

55 
54 
53 
52 
51 
50 

11 
12 
13 
14 

9.659763 
.660009 
.660255 
660501 

4.10 
4.10 
4.10 

9.949170 
.949105 
949040 
.948975 

1.08 
1.08 
1.08 

9.710593 
.710904 
.711215 
.711525 

5.18 
5.18 
5.17 
51ft 

10.289407 
.289096 

.288785 
.288475 

49 

48 
47 
46 

15 
16 

660746 
.660991 

4.08 

.948910 
.948845 

1.08 

.711836 
.712146 

5.17 

.288164 

.287854 

45 
44 

17 

.  661236 

4.08 

948780 

1.08 

.712456 

517 

.287544 

43 

18 
19 
20 

.661481 
.661726 
.661970 

4.08 
4.07 
4.07 

.948715 
948650 
.948584 

1.08 
1.10 
1.08 

.712766 
.713076 
.713386 

5.17 
5.17 
5.17 

.287234 
.286924 
.286614 

42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.662214 
.662459 
.662703 
662946 
.663190 
.663433 
.663677 
.663920 
.664163 
.664406 

4.08 
4.07 
4.05  i 
4.07 
4.05 
4.07 
4.05 
4.05 
4.05 
4.03 

9.948519 
948454 
.948388 
.948323 
.948257 
.948192 
.948126 
.948060 
.947995 
.947929 

1.08 
1.10 
1.08 
1.10 
1.08 
1.10 
1.10 
1.08 
1.10 
1.10 

9.713696 
.714005 
.714314 
.714624 
.714933 
.715242 
.715551 
715860 
.716168 
.716477 

5.15 
5.15 
5.17 
5.15 
5.15 
5.15 
5.15 
5.13 
5.15 
5.13 

10.286304 
.285995 
.285686 
285376 
.285067 
.284758 
.284449 
.284140 
.283832 
.283523 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.664648 
.664891 
.665133 
.665375 
.665617 
.665859 
.666100 
.666342 
.666583 
.666824 

4.05 
4.03 
4.03 
4.03 
4.03 
4  02 
4.03 
4.02 
4.02 
4  02 

9.947863 
.947797 
.947731 
947665 
.947600 
.947533 
.947467 
.947401 
.947&S5 
.947269 

1  10 
1.10 
1.10 
1.08 
1.12 
1.10 
1.10 
1.10 
1.10 
1.10 

9.716785 
.717093 
.717401 
.717709 
.718017 
.718325 
.718633 
.718940 
.719248 
.719555 

5  13 
5.13 
5.13 
5.13 
5.13 
5.13 
5.12 
5.13 
5.12 
5.12 

10.283215 
.282907 
.282599 
.282291 
281983 
.281675 
.281367 
.281060 
.280752 
.280445 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

9.667065 
.667305 
.667546 
.667786 
.668027 
668267 
.668506 
.668746 
.668986 
669225 

4.00 
4.02 
4.00 
4.02 
4.00 
3.98 
4  00 
4.00 
3.98 
3  98 

9  947203 
.947136 
.947070 
.947004 
.946937 
946871 
946804 
.946738 
.946671 
.946604 

1  12 
1.10 
1.10 
1.12 
1.10 
1.12 
1.10 
1.12 
1.12 
1  10 

9.719862 
.720169 
.720476 
.720783 
.721089 
.721396 
.721702 
.722009 
.722315 
.722621 

5.12 
5.12 
5.12 
5.10 
5.12 
5.10 
5.12 
5.10 
5.10 
5.10 

10.280138 
.279831 
.279524 
.279217 
.278911 
.278604 
.278298 
.277991 
.277685 
.277379 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
55 
5G 
57 
58 
59 
60 

9.669464 
669703 
669942 
670181 
670419 
.670658 
670896 
.671134 
.671372 
9.671609 

398 
3.98 
3.98 
3.97 
3.98 
3.97 
3  97 
3.97 
3.95 

9.946538 
.946471 
.946404 
946337 
946270 
946203 
.946136 
.946069 
946002 
9.945935 

1.12 
1.12 
1.12 
1.12 
1.12 
1.12 
1.12 
1.12 
1.12 

9.722927 
723232 
723538 
.723844 
724149 
724454 
724760 
725065 
725370 
9.725674 

5.08 
5.10 
5.10 
5.08 
5.08 
5.10 
5.08 
5.08 
5.07 

10.277073 

.276768 
276462 
27(5156 
275851 
.275546 
275240 
274935 
274630 
10.274326 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  1'. 

Cotang. 

D.I'. 

Tang. 

1 

117° 


COSINES,  TANGENTS,  AND  COTANGENTS. 


161* 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 
3 
4 
5 
6 
7 
8 

9.671609 
.671847 
.672084 
.672321 
.672558 
.672795 
.673032 
.673268 
.673505 

3.97 
3.95 
3.95 
3.95 
3.95 
3.95 
3.93 
3.95 

9.945935 
.  945868 
.945800 
.945733 
.945666 
.945598 
.945531 
.945464 
.945396 

1.12 
1.13 
1.12 
1.12 
1.13 
1.12 
1.12 
1.13 

9.725674 
725979 
!  726284 
.726588 
.726892 
.727197 
.727501 
.727805 
.728109 

5.08 
5.08 
5.07 
5.07 
5.05 
5.07 
5.07 
5.07 

10.274326 
.274021 
.273716 
.273412 
.273108 
.272803 
.272499 
.272195 
.271891 

60 
59 
58 
57 
56 
55 
54 
53 
52 

9 
10 

.673741 
.673977 

3.93 
3.93 

.945328 
.945261 

1.13 
1.12 

.728412 
.728716 

5.05 
5.07 

.271588 
.271284 

51 
50 

3.93 

1.13 

5.07 

11 

9.674213 

9.945193 

9.729020 

10.270980 

49 

12 
13 
14 
15 
16 
17 
18 

.674448 
.674684 
.  674919 
.675155 
.675390 
.675624 
.675859 

3.92 
3.93 
3.92 
3.93 
3.92 
3.90 
3.92 

.945125 
.945058 
.944990 
.944922 
.944854 
.944786 
.944718 

1.13 
1.12 
1.13 
1.13 
1.13 
1.13 
1.13 

.729323 
.729626 
.729929 
.730233 
.730535 
.73083a 
.731141 

5.05 
5.05 
5.05 
5.07 
5.03 
5.05 
5.05 

.270677 
.270374 
.270071 
.269767 
.269465 
.269162 
.268859 

48 
47 
46 
45 
44 
43 
42 

19 
20 

.676094 
.676328 

3.92 
3.90 
3.90 

.944650 
.944582 

1.13 
1.13 
1.13 

.731444 
.731746 

5.05 
5.03 
5.03 

.268556 
.268254 

41 
40 

21 
22 

9.676562 
.676796 

3.90 

9.944514 
.944446 

1.13 

9.732048 
.732351 

5.05 

10.267952 
.267'649 

39 
38 

23 
24 

.677030 
.677264 

3.90 
3.90 

.944377 
.944309 

1.15 
1.13 

.732653 
.732955 

5.03 
5.03 

.267347 
.267045 

37 
36 

25 

.677498 

3.90 

.944241 

1.13 

.733257 

5.03 

.266743 

35 

26 
27 
28 
29 
30 

.677731 
.677964 
.678197 
.678430 
.678663 

3.88 
3.88 
3.88 
3.88 
3.88 
3.87 

.944172 
.944104 
.944036 
.943967 
.943899 

1.15 
1.13 
1.13 
1.15 
1.13 
1.15  i 

.733558 
.733860 
.734162 
.734463 
.734764 

5.02 
5.03 
5.03 
5.02 
5.02 
5.03 

.266442 
.266140 
.265838 
.265537 
.265236 

34 
33 
32 
31 
30 

31 

9.678895 

9.943830 

9.735066 

10.264934 

29 

32 
33 

.679128 
.679360 

3.88 
3.87 

.943761 
.943693 

1.15 
1.13 

.735367 
.735668 

5.02 
5.02 

.264633 
.264&S2 

28 

•27 

34 

.679592 

3.87 

.943624 

1.15 

.735969 

5.02 

.264031 

26 

35 
36 

.679824 
.680056 

3.87 
3.87 

.943555 
.943486 

.15 

.15  | 

.736269 
.736570 

5.00 
5.02 

.263731 
.263430 

25 
24 

37 
38 
39 

.680288 
.680519 
.680750 

3.87 
3.85 
3.85 

.943417 
.943348 
.943279 

.15 
.15 
.15 

.736870 
.737171 
.737471 

5.00 
5.02 
5.00 

.263130 
.262829 
.262529 

23 
22 
21 

40 

.680982 

3.87 

.943210 

•  -15  ' 

.737771 

5.00 

.262229 

20 

3.85 

.15 

5.00 

41 
42 
43 
44 

9.681213 
.681443 
.681674 
.681905 

3.83 
3.85 

3.85 

9.943141 

.942072 
.943003 
.942934 

.15 

.15  i 
.15  : 

•  17 

9.738071 
.738371 
.738671 
.738971 

5.00 
5.00 
5.00 

5  Of) 

10.261929 
.261629 
.261329 
.261029 

19 
18 
17 
16 

45 

46 

47 

.682135 
.682365 
.682595 

3^83 
3.83 

.942864 
.942795 
.942726 

.  .it   ' 

.15 

:  .15  ! 

.739271 
.739570 
.739870 

.uu 
4.98 
5.00 

.260729 
.260430 
.260130 

15 
14 
13 

48 

.682825 

3.83 

.942656 

.17 

.740169 

4.98 

.259831 

12 

49 
50 

.683055 

.683284 

3.83 

3.82 
3.83 

.942587 
.942517 

!  .15  i 
.17 
.15 

.740468 
.740767 

4.  98 
4.98 
4.98 

.259532 
.259233 

11 
10 

51 

52 

9.683514 
.683743 

3.82 

9.942448 

.942378 

.17 

9.741066 
.741365 

4.98 

10.258934 
.258635 

9 

8 

53 

.683972 

3.82 

.942308 

.17 

.741664 

;*j5j| 

.258336 

7 

54 

.684201 

3.82 

.942239 

!  .15 

.741962 

4.97 

.258038 

6 

55 
56 

57 

.684430 
.684658 

.684887 

3.82 
3.80 
3.82 

.942169 
.942099 
.942029 

.17 
.17 

:  .17  ! 

.742261 
.742559 

.742858 

4.98 
4.97 
4.98 

.257739 
.257441 
.257142 

5 

4 
3 

58 

.685115 

3.80 

.941959 

'.  .17 

.743156 

4.97 

4Q7 

.256844 

2 

59 
60 

.685343 
9.685571 

3.80 
3.80 

.941889 
9.941819 

.17 
.17 

.743454 
9.743752 

.VI 

4.97 

.256546 
10.256248 

1 
0 

' 

Cosine. 

D.  1". 

Sine. 

D.  1". 

Cotang. 

D.  r. 

Tang. 

' 

61° 


TABLE    X. — LOGARITHMIC    SINES, 


' 

Sine. 

D.  1". 

Cosine. 

D.  1'. 

Tang. 

D.  1'. 

Cotang. 

' 

0 

1 

2 
3 

9.6R5571 
.685799 
.686027 
.686254 

3.80 

3.80 

3.78 
3  on 

9.941819 
.941749 
.941679 
.941609 

1.17 
1.17 
1.17 

117 

9.743752 
.744050 
.744348 
.744645 

4.07 
4.97 

4-95 

10.256248 
.255950 
.255652 
.255355 

60^ 
59 
58 
57 

4 
5 
6 

7 

.686482 
.686709 
.686936 
.687163 

.oil 

3.78 
3.78 
3.78 

.941539 
.941469 
.941398 
.  941328 

.  1  i 

1.17 
1.18 
1.17 

1  17 

.744943 
.745240 
.745538 
.745835 

4^95 
4.97 
4.95 

.255057 
.254760 
.254462 
.254165 

56 
55 
54 
53 

8 

.687389 

3.77 

3"~ft 

.941258 

1  .  1  1 
1  1ft 

.74ol32 

4.95 

.253868 

52 

9 

.687616 

.  10 

.941187 

1  .  lo 

.746429 

A  n- 

.253571 

51 

10 

.687843 

3.78 
3.77 

.941117 

1  .  17 
1.18 

.  746726 

4  .  Jo 
4.95 

.253274 

50 

11 
12 

9.  688069 
.688295 

3.77 

377 

9.941046 

.940975 

i-i? 

9.747023 
.747319 

4.93 

4  OK 

10.252977 
.252681 

49 

48 

13 
14 
15 

.688521 
.688747 
.688972 

.  <  i 

3.77 
3.75 

.940905 
.940834 
.940763 

1.18 
1.18 

.747616 
.747913 
.748209 

.  yo 
4.95 
4.93 

.252384 
.252087 
.251791 

47 
46 
45 

16 

.689198 

0  .  M   I 

37- 

.940693 

1.17 
11ft 

.748505 

4  no 

.251495 

44 

17 
18 
19 

.689423 
.689648 
.689873 

.  O 

3.75 
3.75 

3r-K 

.940622 
.940551 
.940480 

.  lo 

1.18 
1.18 
11ft 

.748801 
.749097 
.749393 

.yo 
4.93 
4.93 
4  no 

.251199 
.250903 
.250607 

43 
42 
41 

20 

.690098 

.  <•> 

3.75 

.940409 

.  lo 
1.18 

.749689 

.yo 
4.93 

.250311 

40 

21 
22 
23 
24 

9.690323 
.690548 
.690772 
.690996 

3.75 
3.73 
3.73 

3/~q 

9.940338 
.940267 
.940196 
.940125 

1.18 
1.18 
1.18 
1  1ft 

9.749985 
.750281 
.750576 

.750872 

4.93 

4.92 
4.93 
4  92 

10.250015 
.249719 
.249424 
.249128 

39 
38 
37 
36 

25 

.691220 

.  <  O 

.940054 

J  .  lo 
Ion 

.751167 

409 

.248833 

35 

26 
27 

.691444 
.691668 

3  .  73 

3.73 

3IVO 

.939982 
.939911 

.4\) 

1.18 
1  ift 

.751462 
.751757 

.  y/4 
4.92 

.248538 
.248243 

34 
33 

28 
29 
30 

.691892 
.692115 
.692.339 

.  1  •  > 

3.72 
3.73 
3.72 

.939840 
.939768 
.939697 

1  .  lo 
1.20 
1.18 
1.20 

.752052 
.752347 
.752642 

4^92 
4.92 
4.92 

.247948 
.247653 
.247358 

32 
31 

30 

31 
32 
33 

9.692562 
.692785 
.693008 

3.72 
3.72 

3  "9 

9.939625 
.939554 
.939482 

1.18 
1.20 
1  on 

9.752937 
.753231 
.753526 

4.90 
4  92 
4  90 

10.247063 
.246769 
.246474 

29 
28 
27 

34 
35 
36 

.693231 
.693453 
.693676 

.  i  -   \ 

3.70 
3.72  ! 

O  I~A    ! 

.939410 
.939339 
.939267 

1  4\J 

1.18 
1.20 

.753820 
.754115 
.754409 

4^92 
4.90 
4  on 

.246180 

.245885 
.245591 

26 
25 
24 

37 
38 

.693898 
.694120 

3>0 

3TI 

.939195 
.939123 

1.20 

11ft 

.754703 
.754997 

.yu 
4.90 

245297 
.245003 

23 

22 

39 
40 

.694342 
.694564 

.  I  U   ' 

3.70 
3.70 

.939052 
.938980 

.  lo 

1.20 
1.20 

.755291 
.755585 

4^90 
4.88 

.244709 
.244415 

21 
20 

41 

9.694786 

3  PA 

9.938908 

9.755878 

4  on 

10.244122 

19 

42 

43 

44 

.695007 
.695229 
.695450 

.  OO 

3.70 
3  68 

.938836 
.938763 
.938691 

1.28 

1.20 
Ion 

.756172 
.756465 
.756759 

.  JO 

4.88 
.90 

QO 

.243828 
.243535 
.243241 

18 
17 
16 

45 

.695671 

3  68 

.938619 

.4,0 

.757052 

<  .00 

.242948 

15 

46 
47 

.695892 
.696113 

3.08 
3.68 

3  (5ft 

.938547 
.938475 

1  .20 
1.20 

199   ' 

.757345 

.757638 

1  .88 
.88 

.   QQ 

.242655 
.242362 

14 
13 

48 
49 
50 

.696334 
.696554 
.696775 

.OO 

3.67 
3.68 
3.67 

.938402 
.938330 
.938258 

.zx 

1.20 
1.20 
1.22 

.757931 
.758224 
.758517 

•  .  OO 

.88 
4.88 
4.88 

.242069 
.241776 
.241483 

12 
11 

10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.696995 
.697215 
.697435 
.697654 
.697874 
.698094 
.698313 
.698532 
.698751 
9.698970 

3.67 
3.67 
3.65 
3.67 
3.67 
3.65 
3.65 
3.65 
3.65 

9.938185 
.938113 
.938040 
.937967 
.937895 
.937822 
.937749 
.937676 
.937604 
9.937531 

1.20 
1.22 
1.22 
1.20 
1.22 
1.22 
1.22 
1.20 
1.22 

9.758810 
.759102 
.759395 
.759687 
.759979 
.760272 
.760564 
.760856 
.761148 
9.761439 

4.87 
4.88 
4.87 
4.87 
4.88 
4.87 
4.87 
4.87 
4.85 

10.241190 
.240898 
.240605 
240313 
.240021 
.239728 
.239436 
.239144 
.238852 
10.238561 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  1". 

Sine. 

D.  1".  ! 

Cotang. 

D.  1*. 

Tang. 

1 

133 


COSINES,  TANGENTS,  AND  COTANGENTS. 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 
3 

4 
5 
6 

7 
8 

9.698970 
.699189 
.699407 
.699626 
.699844 
.700062 
.700280 
.700498 
.700716 

3.65 
3.63 
3.65 
3.63 
3.63 
3.63 
3.63 
3.63 

3f»9 

9.937531 
.937458 
.937385 
.937312 
.937238 
.937165 
.937092 
.937019 
.936946 

1.22 
1.22 
1.22 
1.23 
1.22 
1.22 
1.22 
.22 
28 

9.761439 
.761731 
.762023 
.762314 
.762606 
.762897 
.763188 
.763479 
.763770 

4.87 
4.87 
4.85 
4.87 
4.85 
4.85 
4.85 
4.85 

10.238561 
.238269 
.237977 
.237686 
.237394 
.237103 
.236812 
.236521 
.236230 

60 
59 
58 
57 
56 
55 
54 
53 
52 

9 

10 

.700^33 
.701151 

.  O/* 

3.63 
3.62 

.936872 
.936799 

'.22 
.23 

.764061 
.764352 

4.85 
4.85 
4.85 

.235939 
.235648 

51 
50 

11 

9.701368 

3  62 

9.936725 

00 

9.764643 

A  CO 

10.235357 

49 

12 
13 

14 

.7J1585 
.701802 
.702019 

3^62 
3.62 
3  62 

.936652 
.936578 
.936505 

^23 

.22 

.764933 
.765224 

.765514 

<*  .00 

4.85 
4.83 

4   Of 

.235067 
.234776 
.234486 

48 
47 
46 

15 
16 
17 
18 

.702236 
.702452 
.702669 

.702885 

3^60 
3.62 
3.60 
3  60 

.936431 
.936357 
.936284 
.936210 

^23 

.22 
.23 
23 

.765805 
.766095 
.766385 
.766675 

.00 

4.83 
4  83 
4.83 

4oq 

.234195 
.233905 
.233615 
.233325 

45 
44 
43 

42 

19 
20 

.703101 
.703317 

3^60 
3.60 

.936136 
.936062 

.'23 
.23 

.766965 
.767255 

.00 

4.83 
4.83 

.233035 
.232745 

41 

40 

21 

9.703533 

3   Of) 

9.935988 

99. 

9.767545 

10.232455 

39 

22 

.703749 

.ou 

3KQ 

.935914 

no 

.767834 

4.82 

40q 

.232166 

38 

23 

.703964 

.Oo 

3  to 

.935840 

.4o 

.768124 

.OO 

.231876 

37 

24 
25 
26 

27 
28 
29 

.704179 
.704395 
.704610 
.704825 
.705040 
.705254 

.00 

3.60 
3.58 
3.58 
3.58 
3.57 

3  to 

.935766 
.935692 
.935618 
.935543 
.935469 
.935395 

'.23 
1.23 
1.25 
1.23 
1.23 

.768414 
.768703 
.768992 
.769281 
.769571 
.769860 

4.83 
4.82 
4.82 
4.82 
4.83 
4.82 

.231586 
.231297 
.231008 
.230719 
.230429 
.230140 

36 
35 
34 
33 
32 
31 

30 

.705469 

.Oo 

3.57 

.935320 

l'.23 

.770148 

4.80 

4.82 

.229852 

30 

31 

32 

9-705683 

.705898 

3.58 
3  57 

9.935246 
.935171 

1.25 

9.770437 
770726 

4.82 

10.229563 
.229274 

29 

28 

33 
34 
35 
36 
37 
38 

.706112 
.706326 
.706539 
.706753 
.706967 
.707180 

3^57 
3.55 
3.57 
3.57 
3.55 

3.  c>r 

.935097 
.935022 
.934948 
.934873 
.934798 
.934723 

1:1 

1.23 
1.25 
1.25 
1.25 
1  .23 

.771015 
.771303 

.771592 
.771880 
.772168 
.772457 

4  .82 
4.80 
4.82 
4.80 
4.80 
4.82 

4QA 

.228985 
.228697 
.228408 
.228120 
.227832 
.227543 

27 
26 
25 
24 
23 
22 

39 

.707393 

OO 

3KK 

.934649 

.772745 

.  OU 
4   Of) 

.227255 

21 

40 

.707606 

.OO 

3.55 

.934574 

IJK 

.773033 

.  oV 

4.80 

.226967 

20 

41 
42 
43 

44 
45 
46 
47 
48 
49 
50 

9.707819 
.708032 
.708245 
.708458 
.708670 
.708882 
.709094 
.709306 
.709518 
.709730 

3.55 
3.55 
3.55 
3.53 
3.53 
3.53 
3.53 
3.53 
3.53 
3.52 

9.934499 
.934424 
.934349 
.934274 
.934199 
.934123 
.934048 
.933973 
.933898 
.933822 

1.25 
1.25 
1.25 
1.25 
1.27 
1.25 
1.25 
1.25 
1.27 
1.25 

9.773321 

.773608 
.773896 
.77418-4 
.774471 
.774759 
.775046 
.775333 
.  775621 
.775908 

4.78 
4.80 
4.80 
4.78 
4.80 
4.78 
4.78 
4.80 
4.78 
4.78 

10.226679 
.226392 
.226104 
.225816 
.225529 
.225241 
.224954 
.224667 
.224379 
.224092 

19 

18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 

9.709941 
.710153 
.710364 

3.53 
3.52 

9.933747 
.933671 
.933596 

1.27 
\'S 

9.776195 

.77(5482 
.776768 

4.78 

4.77 

47ft 

10.22,3805 
.223518 
.223232 

9 

8 

7 

54 
55 
56 
57 
58 
59 
60 

.710575 
.710786 
.710997 
.711208 
.711419 
.711629 
9.711839 

3  .  52 
3.52 
3.52 
3.52 
3.52 
3.50 
3.50 

.933520 
933445 
.933369 
.933293 
.933217 
.933141 

I'M 

1.27 
1.27 
1.27 
1.27 
1.25 

.777055 
.777342 

.777628 
.777915 

.778201 
.778488 
9.778774 

.  to 

4.78 
4.77 
4.78 
4.77 
4.78 
4.77 

.222945 
.222658 
'  .222372 
.222085 
.221799 
.221512 
10.221226 

6 
5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  r.  \ 

1   Sine. 

D.1% 

Cotang. 

D.  r. 

Tang. 

120" 


134 


TABLE    X. — LOGAKITHMIC   SINES, 


' 

Sine. 

D.  r. 

Cosine. 

D.I". 

Tang. 

D.  i". 

Cotang. 

1 

0 

1 

2 
3 
4 
5 
6 

8 
9 

9.711839 
.712050 
.712260 
.712469 
.712679 
.712889 
.713098 
.713308 
.713517 
.713726 

3.52 
3.50 
3.48 
3.50 
3.50 
3.48 
3.50 
3.48 
3.48 
3  48 

9.933066 
.932990 
.632914 
.932838 
.932762 
.932685 
.932609 
.932533 
.932457 
.932380 

1.27  ' 
1.27 
1.27 
1.27 
1.28 
1.27 
1.27 
1.27 
1.23 
1  27 

9.778774 
.779060 
.779346 
.779632 
.779918 
.780203 
.780489 
.780775 
.781060 
.781346 

4.77 
4.77 
4.77 
4.77 
4.75 
4.77 
4.77 
4.75 
4.77 

10.221226 
.220940 
.220654 
.220368 
.220082 
.219797 
.219511 
.219225 
.218940 
.218654 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 

.713935 

3'.48 

.932304 

l!27 

.781631 

4  .75 
4.75 

.218369 

50 

11 

9.714144 

347 

9.932228 

9.781916 

10.218084 

49 

12 
13 
14 
15 

.714352 
.714561 
.714769 
.714978 

.If 

3.48 
3.47 
3.48 
3  47 

.932151 
.032075 
.931998 
.931921 

l!27  j 
1.28  i 
1.28 
1  27 

.782201 

.782486 
.782771 
.783056 

4.75 
4.75 
4.75 
4.75 

4r»K 

.217799 
.217514 
.217229 
.216944 

48 
47 
46 
45 

16 

.715186 

.931845 

1'oQ     j 

.783341 

.  i  •) 

.216659 

44 

17 

18 

.  715394 
.715602 

3.47 
3.47 

34K 

.931763 
.931691 

.>&   i 

1.28  j 
1  28  ' 

.783626 
.783910 

4.75 
4.73 

47K 

.216374 
.216000 

43 

42 

19 
20 

.715809 
.716017 

.40 

3.47 
3.45 

.931614 
.931537 

1^28 
1.28 

.784195 
.784479 

.  1  •) 

4.73 
4.75 

.215805 
.215521 

41 
40 

21 
22 

9.716224 
.716432 

3.47 

9.931460 
.931383 

1.28 

1  9ft 

9.784764 
.785048 

4.73 

10.215236 
.214952 

39 
38 

23 
24 
25 

26 

.716639 
.716846 
.717053 
.717259 

3.45 
3.45 
3.45 
3.43 

.931306 
.931229 
.931152 
.931075 

1  .*0 

1.28 
1.28 
1.28 

1OQ 

.785332 
.785616 
.785900 
.786184 

4.73 
4.73 
4.73 
4.73 

.214668 
.214384 
.214100 
.213816 

37 
36 
35 
34 

27 
28 

.717466 
.717673 

3.45 
3.45 

.930998 
.930921 

.*O 

1.28 
1  30 

.786468 
.786752 

4.73 
4.73 
470 

.213532 
.213248 

33 
32 

29 
30 

.717879 
.718085 

3^43 
3.43 

.930843 
.930766 

l!28 
1.30 

.787036 
.787319 

.  to 
4.72 
4.73 

.212964 
.212681 

31 
30 

31 
32 
33 
34 
35 

9.718291 
.718497 
.718703 
.718909 
.719114 

3.43 
3.43 
3.43 
3.42 

9.930688 
.930611 
.930533 
.930456 
.930378 

1.28 
1.30 
1.28 
1.30 
1  ^in 

9.787603 

.787886 
.788170 
.788453 
.788736 

4.72 
4.73 
4.72 

4.72 

10.212397 
.212114 
.211830 
.211547 
.211264 

29 
28 
27 
26 
25 

36 

.719320 

3.43 

.930300 

1  ,«u 

1  98 

.789019 

4.72 

479 

.210981 

24 

37 

38 

.719525 
.719730 

3.42 
3.42 

.930223 
.930145 

1  .6<J 

1.30 

.789302 
.789585 

.  (A 

4.72 

479 

.210698 
.210415 

23 
22 

39 
40 

.719935 
.7*20140 

3.42 
3.42 
3.42 

.930067 
.929989 

1^30 
1.30 

.789868 
.790151 

.  i  ~ 

4.72 
4.72 

.210132 

.209849 

21 
20 

41 
42 
43 

9.720345 
.720549 
.720754 

3.40 
3.42 

9.929911 
.929833 
.929755 

1  30 
1.30 
1  ^n 

9.790434 
.790716 
.790999 

4.70 
4.72 

47O 

10.209566 
.209284 
.209001 

19 

18 
17 

44 
45 
46 
47 

48 

.720958 
.721162 
.721366 
.72157'0 
.721774 

3.40 
3.40  | 
3.40 
3.40 
3.40 

.929677 
.929599 
.929521 
.929442 
.929364 

1  ,«W 

1  30 
l'30 
1.32 
1.30  ! 

.791281 
.791563 
.791846 
.792128 
.792410 

.  l(J 

4.70 
4.72 
4.70 
4.70 
4.  70 

.208719 

.208437 
.208154 
.207872 
.207590 

16 
15 
14 
13 
12 

49 

.721978 

3.40 

.929286 

QO 

,792692 

47rt 

.207308 

11 

50 

.722181 

3.38 
3.40 

.929207 

liao 

.792974 

.  <u 

4.70 

.207026 

10 

51 
52 
53 

9.722385 

.722588 
.722791 

3.38 
3.38 

9.929129 
.929050 
.928972 

1.32  | 
1.30  ; 
1  32 

9.793256 
.79.3538 
.793819 

4.70 
4.68 

47fl 

10.206744 
.206462 
.206181 

9 

8 

7 

54 
55 

.722994 
.723197 

3.38 
3.38 

.928893 
.928815 

liao  i 

1  S2  I 

.794101 
.794383 

.  «u 

4.70 
4  68 

.205899 
.205617 

6 
5 

56 

57 

.723400 
.723603 

3.38 
3.38 

.928736 
.928657 

1.O/6   I 

1.32  | 
1  3° 

.794664 
.794916 

4>0 
4  68 

.205336 
.205054 

4 

3 

58 
59 
60 

.723805 
.724007 
9.724210 

3.37 
3.37 
3.38 

.928578 
.928499 
9.928420 

j  .  <j/£  j 
1.32 
1.32 

.795227 
.795508 
9.  79578  J 

4l68 
4.68 

.204773 
.204492 
10.204211 

2 
1 
0 

' 

Cosine. 

D.  r. 

Sine. 

D.  r. 

Cotang. 

D.I". 

Tang. 

' 

121° 


135 


COSINES,   TANGENTS,   AND  COTANGENTS. 


' 

Sine. 

D.  1'. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

9.724210 

9.928420 

9.795789 

4  Aft 

10.204211 

60 

1 

.724412 

0  Olt 

.928342 

^  00 

.796070 

.Do 

.203930 

59 

2 

.724614 

o.o7 

3°.7 

.928263 

1  .0^ 
Iqq 

.796351 

4  Aft 

.203649 

58 

3 

.724816 

.of 

3  f)K 

.988183 

.00 

.796632 

.Do 
4  Aft 

.203368 

57 

4 
5 

.725017 
.725219 

.OO 

3.37 

3rtK 

.928104 
.928025 

l!32 
Iqo 

.796913 
.797194 

.Do 

4.68 

4A7 

.203087 
.202806 

56 
55 

6 

.725420 

,OO 

3q7 

.927946 

,tS6 

Ion 

.797474 

.D< 

4  Aft 

.202526 

54 

7 

.725622 

.Of 
3  OK 

.927867 

.04 
Iqq 

.797755 

4.  DO 

A  CO 

.202245 

53 

8 

.725823 

.00 
3qe 

.927787 

.00 
109 

.798036 

J;g     .201964 

52 

9 

10 

.726024 
.726225 

.  oO 

3.35 
3.35 

.927708 
.927623 

,WB 

1.32 
1.33 

.798316 
.798596 

4^67 
4.68 

.201684 
.201404 

51 
50 

11 

9.726426 

3qq 

9.927549 

9.798877 

4A7 

10.201123 

49 

12 

13 

.726626 
.726827 

.OO 

3.35 

3qq 

.927470 
.927390 

1^33 

Iqq 

.799157 
.799437 

.D< 

4.67 
4  67 

.200843 
.200563 

48 
47 

U 
15 
16 

.72i'027 
.727228 
.727428 

.  OO 

3.35 
3.33 

.927310 
.927231 
.927151 

.  oo 

1.32 
1.33 

.799717 
.799997 
.800277 

4.67 
4.67 

.200283 
.200003 
.199723 

46 
45 
44 

17 

.727628 

3  33 

3qq 

.927071 

1  JM 

.800557 

4.67 

4CK 

.199443 

43 

18 
19 
20 

.727828 
.728027 
.728227 

.OO 

3.32 
3.33 
3.33 

.926991 
.926911 
.926831 

lias 

1.38 

1.33 

.800836 
.801116 
.801396 

.DO 

4.67 
4.67 
4.65 

.199164 

.198884 
.198604 

42 

41 
40 

21 

9.728427 

3qo 

9.926751 

Iqq 

9.801675 

10.198325 

39 

22 

.728626 

.06 

.926671 

.00 

.801955 

4.O< 

.198045 

38 

23 

.728825 

3.32 
309 

.926591 

1.33 
Iqq 

.802234 

4.65 

4  OK 

.197766 

37 

24 
25 
26 

.729024 
.729223 

.729422 

.Ow 

3.32 
3.32  * 

.926511 
.926431 
.926351 

.  oo 

1.33 
1.33 

.802513 
.802792 
.803072 

.DO 

4.65 
4.67 

.197487 
.197208 
.196928 

36 
35 
34 

27 

28 

.729621 
.729820 

3.  '32 

3Qft 

.926270 
.926190 

I'M 

Iqq 

.803351 
.803630 

4^65 

4(*K 

.196649 
.196370 

33 

32 

29 
30 

.730018 
.730217 

.oU 

3.32 
3.30 

.926110 
.926029 

.00 

1.35 
1.33 

.803909 
.804187 

.DO 

4.63 
4.65 

.196091 
.195813 

31 
30 

31 
32 

9  ,730415 
.730613 

3.30 

3  Of) 

9.925949 

.925868 

1.35 

Iqq 

9.804466 
.804745 

4.65 

10.195534 
.195255 

29 

28 

33 

34 

.730811 
.731009 

.oU 

3.30 

.925788 
.925707 

.OO 

1.35 

IOK 

.805023 
.805302 

4  .  63 
4  (55 

.194977 
.194698 

27 
26 

35 

36 

.731206 
.73J404 

3^30 

3qr» 

.925626 
.925545 

.00 

1.35 

.805580 
.805859 

4^65 

.194420 
.194141 

25 
24 

37 
38 
39 

.731602 
.731799 
.731996 

.oU 

3.28 
3.28 
390 

.925465 
.925384 
.925303 

l!35 
1.35 

.806137 
.806415 
.806693 

4.63 
4.63 

.193863 
.193585 
.193307 

23 
22 
21 

40 

.732193 

.Ao 

3.28 

.925222 

1J6 

.806971 

4.  '63 

.193029 

20 

41 

9.732390 

9.925141 

9.807249 

4A°. 

10.192751 

19 

42 
43 
44 
45 

.732587 
.732784 
.732980 
.733177 

3.  '28 
3.27 
3.28 

397 

.925060 
.924979 
.924897 
.924816 

•('.35 
1-37 
1.35 

IOK 

.807527 
.807805 
.808083 
.808361 

.Do 

4.63 
4.63 
4.63 

.192473 
.192195 
.191917 
.191639 

IS 
17 
16 
15 

46 

47 
48 

.733373 
.733569 
.733765 

.iff 

3.27 
3.27 

397 

.924735 
.924654 
.924572 

.oO 

1.35 
1.37 

.808638 
.808916 
.809193 

4^63 
4.62 

.191362 
.191084 
.190807 

14 
13 
12 

49 
50 

.733961 
.734157 

.*f 

3.27 
3.27 

.924491 
.924409 

1.37 
1.35 

.809471 
.809748 

4^62 
4.62 

.190529 
.190252 

11 
10 

51 
52 
53 

9.734353 
.734549 
.734744 

3.27 
3.25 

9.924328 
.924246 
.924164 

1.37 
1.37 

9.810025 
.810302 
.810580 

4.62 
4.63 

10.189975 
.189698 
.189420 

9 

8 

7 

54 
55 
56 
57 
58 
59 
60 

.734939 
.735135 
.735330 
.735525 
.735719 
.735914 
9.736109 

3.25 
3.27 
3.25 
3.25 
3.23 
3.25 
3.25 

.924083 
.924001 
.923919 
.9213837 
.923755 
.923673 
9-923591 

1.35 
1.37 
1.37 
1.37 
1.37 
1.37 
1.37 

.810&57 
.811134 
.811410 
.811687 
.811964 
.812241 
9.812517 

4.62 
4.62 
4.60 
4.62 
4.62 
4.62 
4.60 

.189143 
.188866 
.188590 
.188313 
.188036 
.187759 
10.187483 

6 
5 
4 
3 
2 
1 
0 

'   Cosine. 

D.  1". 

Sine. 

D.  1*. 

Cotang.  D.  1'. 

Tang. 

' 

136 


57- 


83* 


TABLE    X. — LOGARITHMIC    SINES, 


' 

Sine. 

D.  r. 

Cosine. 

D.I'. 

Tang. 

D.  1'. 

Cotang. 

' 

0 

9.736109 

3  no 

9.923591 

1  Vf 

9.812517 

4A9 

10.187483 

60 

1 

2 
3 
4 

.736303 
.736498 
.736692 
.736886 

.  -CO 

3.25 
3.23 
3.23 

3f)Q 

.923509 
.923427 
.923345 
.923263 

i  .04 
1.37 
.37 
.37 

07 

.812794 
.813070 
.813347 
.813623 

.D* 

4.60 
4.62 
4.60 

.187206 
.186930 
.186653 
.186377 

59 
58 
57 
56 

5 

.737080 

.•"JO 

.923181 

.o< 

.813899 

4.60 

.186101 

55 

6 

7 

.737274 

.737467 

3.23 

3.22 
390 

.923098 
.923016 

.38 
.37 

00 

.814176 
.814452 

4.62 
4.60 

.185824 
.185548 

54 
53 

8 

.737661 

,*o 

.922933 

.00 

.814728 

4  fin 

.185272 

52 

9 

.737855 

3.23 

.  92285  I 

'  QO 

.815004 

.ou 

.184996 

51 

10 

.738048 

3.22 
3.22 

.922768 

.OO 

1.37 

.815280 

4.60 
4.58 

.184720 

50 

11 
12 

9.738241 
.738434 

3.22 

9.922686 
.922603 

1.38 

9.815555 

.815831 

4.60 
4  fin 

10.184445 
.184169 

49 

48 

13 
14 

.738627 
.738820 

3^22 

.922520 
.922438 

1.91 

.816107 
.816382 

.DU 

4.58 

.183893 
.183618 

47 
46 

15 

.739013 

3.22 

.922355 

1  .38 

.816658 

4.  60 

.18a342 

45 

16 

.739206 

3.22 

.922272 

1  .38 

.816933 

4.58 

.183067 

44 

17 

.739398 

3.20 

.922189 

'oo 

.817209 

A   tO 

.182791 

43 

18 

.739590 

3.20 

.922106 

1  .08 

1QQ 

.817484 

4.58 

4KQ 

.182516 

42 

19 

.739783 

3.22 

.922023 

.00 

.817759 

.00 

.182241 

41 

20 

.739975 

3.20 
3.20 

.921940 

1.38 
1.38 

.818035 

4.60 
4.58 

.181965 

40 

21 
22 
23 
24 

9.740167 
.740359 
.740550 
.740742 

3.20 
3.18 
3.20 

9.921857 
.921774 
.921691 
.921607 

1.38 
1.38 
1.40 

9.818310 
.818585 
.818860 
.819135 

4.58 
4.58 
4.58 

10.181690 
.181415 
.181140 
-  .180865 

39 
38 
37 
36 

25 

.740934 

3.20 

.921524 

1.38 

.819410 

4.58 

.180590 

35 

26 
27 
28 
29 

.741125 
.741316 
.741508 
.741699 

3.18 
3.18 
3.20 
3.18 

.921441 
.921357 
.921274 
.921190 

l'.40 
1.38 
1.40 

.819684 
.819959 
.820234 
.820508 

4.57 
4.58 
4.58 
4.57 

.180316 
.180041 
.179766 
.179492 

34 
33 
32 
31 

30 

.741889 

3.17 
3.18 

.921107 

1^40 

.820783 

4!57 

.179217 

30 

31 
32 

9.742080 
.742271 

3.18 

9.921023 
.920939 

.40 

QQ 

9.821057 
.821332 

4.58 

10.178943 
.178668 

29 
28 

33 
34 

.742462 
.742652 

3.18 
3.17 

.920856 
.920772 

.OO 

.40 

.821606 
.821880 

4^57 

.178394 
.178120 

27 
26 

35 
36 

.742842 
.743033 

3.17 
3.18 

.920688 
.920604 

.40 
.40 

.822154 

.822429 

4.57 

4.58 

.177846 
.177571 

25 
24 

37 

.743223 

3.17 

.920520 

.40 

.822703 

7*5 

.177297 

23 

38 
39 

40 

.743413 
.743602 
.743792 

3.17 
3.15 
3.17 
3.17 

.920436 
.920352 
.920268 

.40 
.40 
.40 
.40 

.822977 
.823251 
.823524 

4.5< 
4.57 
4.55 
4.57 

.177023 
.176749 
.176476 

22 
21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

9.743982 
.744171 
.744361 
.744550 
.744739 
.744928 
.745117 
.745306 
.745494 
.745683 

3.15 
3.17 
3.15 
3.15 
3.15 
3.15 
3.15 
3.13 
3.15 
3.13 

9.920184 
.920099 
.920015 
.919931 
.919846 
.919762 
.919677 
.919593 
.919508 
.919424 

.42 
.40 
.40 
.42 
.40 
.42 
.40 
.42 
1.40 
1.42 

9.823798 
.824072 
.824345 
.824619 
.824893 
.825166 
.825439 
.825713 
.825986 
.826259 

4.57 
4.55 
4.57 
4.57 
4.55 
4.55 
4.57 
4.55 
4.55 
4.55 

10.176202 
.175928 
.175655 
.175381 
.175107 
.174834 
.174561 
.174287 
.174014 
.173741 

19 

18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

9.745871 
.746060 
.746248 
.746436 
.746624 
.746812 
.746999 
.747187 
.747374 
9.747562 

3.15 
3.13 
3.13 
3.13 
3.13 
3.12 
3.13 
3.12 
3.13 

9.919339 
.919254 
.919169 
.919085 
.919000 
.918915 
.918830 
.918745 
.918659 
9.918574 

1.42 
1.42 
1.40 
1.42 
1.42 
1.42 
1.42 
1.43 
1.42 

9.826532 
.826805 
.827078 
.827351 
.827624 
.827897 
.828170 
.828442 
.828715 
9.828987 

4.55 
4.55 
4.55 
4.55 
4.55 
4.55 
4.53 
4.55 
4.53 

10.173468 
.173195 
.172922 
.172649 
.172376 
.172103 
.171830 
.171558 
.171285 
10.  171013 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

~ 

Cosine. 

D.  1'. 

Sine. 

'  D.  1'. 

Cotang. 

D.  1". 

Tang. 

' 

123° 


137 


34* 


COSINES,  TANGENTS,  AND  COTANGENTS. 


145* 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 
3 
4 
5 

6 

8 
9 

9.747562 

.747749 
.747936 
.748123 
.748310 
.748497 
.748683 
.748870 
.749056 
.749243 

3.12 
3.12 
3.12 
3.12 
3.12 
3.10 
3.12 
3.10 
3.12 

9.918574 
.918489 
.918404 
.918318 
.918233 
.918147 
.918062 
.917976 
.917891 
.917805 

1.42 
1.42 
.43 
.42 
.43 
.42 
.43 
.42 
.43 

9.828987 
.829260 
.829532 
.829805 
.830077 
.830349 
.830621 
.830893 
.831165 
.831437 

4.55 
4.53 
4.55 
4.53 
4.53 
4.53 
4.53 
4.53 
4.53 

10.171013 
.170740 
.170468 
.170195 
.169923 
.169651 
.169379 
.169107 
.168835 
.168563 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

10 

.749429 

3.10 
3.10 

.917719 

.43 
.42 

.831709 

4.53 
4.53 

.168291 

50 

11 
12 

9.749615 
.749801 

3.10 

31ft 

9.917634 
.917548 

.43 

XQ 

9.831981 
.832253 

4.53 

4eq 

10.168019 
.167747 

49 

48 

13 
14 
15 
16 

.749987 
750172 
!  750358 
.750543 

.  1U 

3.08 
3.10 
3.08 

.917462 
.917376 
.917290 
.917204 

.40 

.43 
.43 
.43 

.832525 
.832796 
.833068 
.833339 

.Oj 

4.52 
4.53 
4.52 

.167475 
.167204 
.166932 
.166661 

47 
46 
45 
44 

17 
18 

.750729 
.750914 

3.10 
3.08 

.917118 
.917032 

.43 
.43 

.833611 

.833882 

4.53 

4.52 

.166389 
.166118 

43 
42 

19 
20 

.751099 
.751284 

3.08 
3.08 
3.08 

.916946 
.916859 

.43 
.45 
.43 

.834154 
.834425 

4.53 

4.52 
4.52 

.165846 
.165575 

41 
40 

21 

9.751469 

3AO 

9.916773 

9.834696 

10.165304 

39 

22 

.751654 

.UC 

.916687 

.4o 

.834967 

4.52 

.165033 

38 

23 
24 
25 
26 
27 

.751839 
.752023 
.752208 
.752392 
.752576 

3.08 
3.07 
3.08 
3.07 
3.07 

.916600 
.916514 
.916427 
.916341 
.916254 

.45 
.43 
.45 
.43 

.45 

.835238 
.835509 
.835780 
.836051 
.836322 

4.52 
4.52 
4.52 
4.52 
4.52 

.164762 
.164491 
.164220 
.163949 
.  163678 

37 
36 
35 
34 
33 

28 

.752760 

3.07 

.916167 

.45 

.836593 

4.52 

.163407 

32 

29 

.752944 

3.07 

.916081 

.43 

.836864 

4.52 

.163136 

31 

30 

.753128 

3.07 

.915994 

.45 

.837134 

4.50 

.162866 

30 

3.07 

.45 

4.52 

31 

32 

9.753312 
.753495 

3.05 

9.915907 
.915820 

•45 

9.837405 

.837675 

4.50 

10.162595 
.162325 

29 

28 

33 

.753679 

3.07 

.915733 

.45 

.837946 

4.52 

.162054 

27 

34 
35 

36 
37 

.753862 
.754046 
.754229 
.754412 

3.07 
3.07 
3.05 
3.05 

.915646 
.915559 
.915472 
.915385 

.45 
.45 
.45 
.45 

.838216 
.838487 
.838757 
.839027 

4.50 
4.52 
4.50 
4.50 

.161784 
.161513 
.161243 
.160973 

26 
25 
24 
23 

38 
39 
40 

.754595 
.754778 
.754960 

3.05 
3.05 
3.03 

.915297 
.915210 
.915123 

.47 
.45 
1.45 

.839297 
.839568 
.839838 

4.50 
4.52 
4.50 

.160703 
.160432 
.160162 

22 
21 
20 

3.05 

1.47 

4.50 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

9.755143 
.755326 
.755508 
.755690 

.755872 
.756054 
.756236 
.756418 
.756600 
.756782 

3.05 
3.03 
3.03 
3.03 
3.03 
3.03 
3.03 
3.03 
3.03 
3.02 

9.915035 
.914948 
.914860 
.914773 
.914685 
.914598 
.914510 
.914422 
.914334 
.914246 

1.45 
1.47 
1.45 
1.47 
1.45 
1.47 
1.47 
1.47 
1.47 
1.47 

9.840108 
.840378 
.840648 
.U0917 
.841187 
.841457 
.841727 
.841996 
.842266 
.842535 

4.50 
4.50 
4.48 
4.50 
4.50 
4.50 
4.48 
4.50 
4.48 
4.50 

10.159892 
.159622 
.159352 
.159083 
458813 
.158543 
.158273 
.158004 
.157734 
.157465 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 
53 
54 

55 
56 

9.756963 
.757144 
.757326 
.757507 

.757688 
.757869 

3.02 
3  03 
3.02 
3.02 
3.02 

3  rig 

9.914158 
.914070 
.913982 
.913894 
.913806 
.913718 

1.47 
1.47 
1.47 
1.47 
.47 

9.842805 
.843074 
.843343 
.843612 
.843882 
.844151 

4.48 
4.48 
4.48 
4.50 
4.48 

10.157195 
156926 
.156657 
.156388 
.156118 
.155849 

9 

8 
7 
6 
5 
4 

57 
58 
59 
60 

.758050 
.758230 
.758411 
9.758591 

.\Jfii 

3.00 
3.02 
3.00 

.913630 
913541 
.913453 
9.913365 

!  .47 
.48 
.47 
.47 

.844420 
.844689 
.844958 
9.845227 

4.48 
4.48 
4.48 
4.48 

.155580 
455311 
455042 
10454773 

3 
2 
1 
0 

' 

Cosine. 

D.  1". 

Sine. 

D.  1". 

Cotang. 

D.  1'. 

Tang. 

' 

134° 


55* 


35' 


TABLE    X. — LOGARITHMIC   SINES, 


' 

Sine. 

D.  1'. 

Cosine. 

D.  1". 

Tang. 

D.  r. 

Cotang. 

• 

0 

1 

2 
3 

9.758591 

.758772 
.758952 
.759132 

3.02 
3.00 

3.00 
300 

9.913365 
.913270 
.913187 
.913099 

1.48 
1.48 
1.47 
1  AA 

9.845227 
.845496 
.845764 
.846033 

4.48 
4.47 

4.48 

10.154773 
.154504 
.154236 
.153967 

60 
59 
58 
57 

4 

.759312 

.00 

.913010 

1  .4o 

.846302 

4.48 

.153698 

56 

5 
6 
7 
8 
9 
10 

.759492 
.759672 
.759852 
.760031 
.760211 
.760390 

3.00 
3.00 
3.00 
2.98 
3.00 
2.98 
2.98 

.912922 
.912833 
.912744 
.912655 
.912566 
.912477 

1.47 
1.48 
1.48 
1.48 
1.48 
1.48 
1.48 

.846570 
.846839 
.847108 
.847376 
.847644 
.847913 

4.47 
,  4.48 
4.48 
4.47 
4.47 
4.48 
4.47 

.153430 
.153161 
.152892 
.152624 
.152356 
.152087 

55 
54 
53 
52 
51 
50 

11 

9.760569 

9.912388 

9.848181 

10.151819 

49 

12 
13 
14 
15 

.760748 
.760927 
.761106 
.761285 

2.98 
2.98 
2.98 
2.98 

.912299 
.912210 
.912121 
.912031 

1  .48 
1.48 
1.48 
1.50 

.848449 
.848717 
.848986 
.849254 

4.47 
4.47 
4.48 
4.47 

.151551 
.151283 
.151014 
.150746 

48 
47 
46 
45 

16 
17 

.761464 
.761642 

2.98 
2.97 

.911942 
.911853 

1.48 
1.48 

.849522 
.849790 

4.47 
4.47 

.150478 
.150210 

44 
43 

18 
19 
20 

.761821 
.761999 
.762177 

2.98 
2.97 
2.97 
2.98 

.911763 
.911674 
.911584 

1.50 
1.48 
1.50 
1.48 

.850057 
.850325 
.850593 

4.45 
4.47 
4.47 
4.47 

.149943 
.149675 
.149407 

42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.762356 
.762534 
.762712 
.762889 
.763067 
.763245 
.763422 
.763600 
.763777 
.763954 

2.97 
2.97 
2.95 
2.97 
2.97 
2.95 
2.97 
2.95 
2.95 
2.95 

9.911495 
.911405 
.911315 
.911226 
.911136 
.911046 
.910956 
.910866 
.910776 
.910686 

1.50 
1.50 
1.48 
1.50 
1.50 
1.50 
1.50 
1.50 
1.50 
1.50 

9.850861 
.851129 
.851396 
.851664 
.851931 
.852199 
.852466 
.852733 
.853001 
.853268 

4.47 
4.45 
4.47 
4.45 
4.47 
4.45 
4.45 
4.47 
4.45 
4.45 

10.149139 
.148871 
.148604 
.148336 
.148069 
.147801 
.147534 
.147267 
.146999 
.146732 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 

9.764131 
.764308 
.764485 

2.95 
2.95 

9.910596 
.910506 
.910415 

1.50 
1.52 

9.853535 

.853802 
.854069 

4.45 
4.45 

10.146465 
.146198 
.145931 

29 
28 
27 

34 

.764662 

2.95 

.910325 

1  .50 

.854336 

4.45 

.145664 

26 

35 
36 
37 
38 
39 
40 

.764838 
.765015 
.765191 
.765367 
.765544 
.765720 

2.93 
2.95 
2.93 
2.93 
2.95 
2.93 
2.93 

.910235 
.910144 
.910054 
.909963 
.909873 
.909782 

1.50 
1.52 
1.50 
1.52 
1.50 
1.52 
1.52 

.854603 
.854870 
.855137 
.855404 
.855671 
.855938 

4.45 
4.45 
4.45 
4.45 
4.45 
4.45 
4.43 

.145397 
.145130 
.144863 
.144596 
.144329 
.144062 

25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 

9.765896 
.766072 
.766247 
.766423 
.766598 

2.93 
2.92 
2.93 
2.92 

9.909691 
.909601 
.909510 
.909419 
.909328 

1.50 
1.52 
1.52 
1.52 

9.856204 
.856471 
.856737 
.857004 
.857270 

4.45 
4.43 
4.45 
4.43 

10.143796 
.143529 
.143263 
.142996 
.142730 

19 
18 
17 
16 
15 

46 
47 
48 
49 
50 

.766774 
.766949 
.767124 
.767300 
.767475 

2.93 
2.92 
2.92 
2.93 
2.92 
2.90 

.909237 
.909146 
.909055 
.908964 
.908873 

1  .52 
1.52 
1.52 
1.52 
1.52 
1.53 

.857537 
.857803 
.858069 
.858336 
.858602 

4  .45 
4.43 
4.43 
4.45 
4.43 
4.43 

.142463 
.142197 
.141931 
.141664 
.141398 

14 
13 
12 
11 
10 

51 

9.767649 

9.908781 

9.858868 

10.141132 

9 

52 
53 
54 

.767824 
.767999 
.768173 

2.92 
2.92 
2.90 

.908690 
.908599 
.908507 

l!52 
1.53 

.859134 
.859400 
.859666 

4^43 
4.43 

.140866 
.140600 
.140334 

8 
7 
6 

55 

56 
57 
58 
59 
60 

.768348 
.768522 
.768697 
.768871 
.769045 
9.769219 

2.92 
2.90 
2.92 
2.90 
2.90 
2.90 

.908416 
.908324 
.908233 
.908141 
.908049 
9.907958 

1  .52 
1.53 
1.52 
1.53 
1.53 
1.52 

.859932 
.860198 
.860464 
.860730 
.860995 
9.861261 

4.  '43 
4.43 
4.43 
4.42 
4.43 

.140068 
.139802 
.139536 
.139270 
.139005 
10.138739 

5 
4 
3 
2 
1 
0 

' 

Cosine. 

D.  1'. 

Sine. 

D.  r.  ! 

Cotang. 

D.  1". 

Tang. 

' 

125° 


139 


COSINES,  TANGENTS,   AND  COTANGENTS. 


143' 


' 

Sine. 

D.  1". 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 
3 

9.769219 
.769393 
.769566 
.769740 

2.90 

2.88 
2.90 

2QQ 

9.907958 
.907866 
.907774 
.907682 

1.53 
1.53 
1.53 
1  53 

9.861261 
.861527 
.861792 
.862058 

4.43 
4.42 
4.43 

10.138739 

.138473 
.138208 
.137942 

60 

59 
58  < 
57 

4 

5 

.769913 

.770087 

.OO 

2.90 

.907590 
.907498 

1^53 

.862323 

.862589 

4  '.43 

.137677 
.137411 

56 
55 

6 

8 
9 
10 

.770260 
.770433 
.770606 
.770779 
.770952 

2.88 
2.88 
2.88 
2.88 
2.88 
2.88 

.907406 
.907314 
.907222 
.907129 
.907037 

1  .53 
1.53 
1.53 
1.55 
1.53 
1.53 

.862854 
.863119 
.863385 
.863650 
.863915 

4.42 
4.42 
4.43 
4.42 
4.42 
4.42 

.137146 
.136881 
.136615 
.136350 
.136085 

54 
53 
52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 

9.771125 
.771298 
.771470 
.771643 
.771815 
.771987 
.772159 
.772331 
.772503 

2.88 
2.87 
2.88 
2.87 
2.87 
2.87 
2.87 
2.87 

9.906945 
.906852 
.906760 
.906667 
.906575 
.906482 
.906389 
.906296 
.906204 

1.55 
1.53 
.55 
.53 
.55 
.55 
.55 
.53 

9.864180 
.864445 
.864710 
.864975 
.865240 
.865505 
.865770 
.866035 
.866300 

4.42 
4.42 
4.42 
4^42 
4.42 
4.42 
4.42 
4.42 

10.135820 
.135555 
.135290 
.135025 
.134760 
.134495 
.134230 
.133965 
.133700 

49 
48 
47 
46 
45 
44 
43 
42 
41 

20 

.772675 

2.87 
2.87 

.906111 

.55 
.55 

.866564 

4.40 
4.42 

.133436 

40 

21 

9.772847 

9.906018 

9.866829 

10.133171 

39 

22 
23 
24 
25 

.773018 
.773190 
.773361 
.773533 

2.85 
2.87 
2.85 
2.87 

2QK 

.905925 
.905832 
.905739 
.905645 

.55 
.55 
.55 
.57 

.867094 
.867358 
.867623 

.867887 

4.42 
4.40 
4.42 
4.40 

.  132906 
.132642 
.132377 
.132113 

38  ; 
37 
36 
35 

26 
27 

.773704 

.773875 

.00 
2.85 

2QK 

.905552 
.905459 

.55 
.55 

.868152 
.868416 

4.42 
4.40 

.131848 
.131584 

34 
33 

28 
29 
30 

.774046 
.774217 
.774388 

.00 
2.85 
2.85 
2.83 

.905366 
.905272 
.905179 

1.55 
1.57 
1.55 
1.57 

.868680 
.868945 
.869209 

4.40 
4.42 
4.40 
4.40 

.131320 
.131055 
.130791 

32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.774558 
.774729 
.774899 
.775070 
.775240 
.775410 
.775580 
.775750 
.775920 
.776090 

2.85 
2.83 

2.85 
2.83 
2.83 
2.83 
2.83 
2.83 
2.83 
2.82 

9.905085 
.904992 
.904898 
.904804 
.904711 
.904617 
.904523 
.904429 
.904335 
.904241 

1.55 
1.57 
1.57 
1.55 
1.57 
1.57 
1.57 
1.57 
1.57 
1.57 

9.869473 
.869737 
.870001 
.870265 
.870529 
.870793 
.871057 
.871321 
.871585 
.871849 

4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.40 
4.38 

10.130527 
.130263 
.129999 
.129735 
.129471 
.129207 
.128943 
.128679 
.128415 
.128151 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 

9.776259 
.776429 

.776598 

2.83 

2.82 

2QQ 

9.904147 
.904053 
.903959 

1.57 
1.57 

9.872112 
.872376 
.872640 

4.40 
4.40 

10.127888 
.  127624 
.127360 

19 
18 
17 

44 

45 
46 
47 
48 
49 
50 

.776768 
.776937 
.777106 
.777275 
.777444 
.777613 
.777781 

.  OO 

2.82 
2.82 
2.82 
2.82 
2.82 
2.80 
2.82 

.903864 
.903770 
.903676 
.903581 
.903487 
.903392 
.903298 

1^57 
1.57 
1.58 
1.57 
1.58 
1.57 
1.58 

.872903 
.873167 
.873430 
.873694 
.873957 
.874220 
.874484 

4.38 
4.40 
4.38 
4.40 
4.38 
4.38 
4.40 
4.38 

.127097 
.126833 
.126570 
.126306 
.126043 
.125780 
.125516 

16 
15 
14 
13 
12 
11 
10 

51 

9.777950 

9.903203 

9.874747 

10.125253 

9 

52 

.778119 

on 

.903108 

1.58 

.875010 

4.38 

.124990 

8 

53 
54 

.778287 
.778455 

2^80 

.903014 
.902919 

1  .57 

1.58 

•  to 

.875273 

.875537 

4.38 
4.40 

4QQ 

.124727 
.124463 

7 

6 

55 

.778624 

o  en 

.902824 

.  .Do 

•   f-Q 

.875800 

.00 

.124200 

5 

56 

.778792 

«'oA 

.902729 

.Do 

.876063 

4.38 

.123937 

4 

57 

.778960 

2.80 

.902634 

!  .58 

.876326 

4.38 

.123674 

3 

58 

.779128 

2.80 

.902539 

.58 

.876589 

4.38 

.123411 

2 

59 

.779295 

2.78 

.902444 

'.  .58 

.876852 

4.38 

.123148 

1 

60 

9.779463 

2.80 

9.902349 

'.  .58 

9.877114 

4.37 

! 

10.122886 

0 

' 

Cosine. 

D.  1". 

Sine. 

D.I". 

Cotang. 

D.  r.  1 

Tang. 

> 

126° 


53« 


37° 


TABLE   X. — LOGARITHMIC    SINES, 


' 

Sine. 

D.  1".  i 

i 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

> 

0 

1 

2 

3 
4 
5 

9.779463 
.779631 
.779798 
.779966 
.780133 
.780300 

2.80 
2.78 
2.80 
2.78 
2.78 

27ft 

9.902349 
.902253 
.902158 
.902063 
.901967 
.901872 

1.60 
1.58 
1.58 
1.60 
1.58 
1  fin 

9.S77114 
.87.7377 
.877640 
.877903 
.878165 
.878428 

4.38 
4.38 
4.38 
4.37 
4.38 

4  Oft 

10.122886 
.122623 
.122360 
.122097 
.121835 
.121572 

60 
59 
58 
57 
56 
55 

6 

7 

9 

.780467 
.780634 
.780801 
.780968 

.  to 

2.78 
2.78 
2.78 
2  77 

.901776 
.901681 
.901585 
.901490 

1  .OU 

1.58 
1.60 
1.58 

.878691 
.878953 
.879216 
.879478 

.00 

4.37 
4.38 
4.37 

4OQ 

.121309 
.121047 
.120784 
.120522 

54 
53 
52 
51 

10 

.781134 

2.78 

.901394 

I'M 

.879741 

.OO 

4.37 

.120259 

50 

11 
12 

9.781301 
.781468 

2.78 

277 

9.901298 
.901202 

1.60 

9.880003 
.880265 

4.37 
400 

10.119997 
.119735 

49 

48 

13 
14 
15 
16 
17 
18 
19 
20 

.781634 
.781800 
.781966 
.782132 
.782298 
.782464 
.782630 
.782796 

.  i  I 

2.77 
2.77 
2.77 
2.77 
2.77 
2.77 
2.77 
2.75 

.901106 
.901010 
.900914 
.900818 
.900722 
.900626 
.900529 
.900433 

l!60 
1.60 
1.60 
1.60 
1.60 
1.62 
1.60 
1.60 

.880528 
.880790 
.881052 
.881314 
.881577 
.881839 
.882101 
.882363 

.00 
4.37 
4.37 
4.37 
4.38 
4.37 
4.37 
4.37 
4.37 

.119472 
.119210 
.118948 
.118686 
.118423 
.118161 
.117899 
.117637 

47 
46 
45 
44 
43 
42 
41 
40 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 

9.782961 
.783127 
.783292 
.783458 
.783623 
.783788 
.783953 
.784118 
.784282 
.784447 

2.77 
2.75 
2.77 
2.75 
2.75 
2.75 
2.75 
2.73 
2.75 
2.75 

9.900337 
.900240 
.900144 
.900047 
.899951 
.899854 
.899757 
.899660 
.899564 
.899467 

1.62 
1.60 
1.62 
1.60 
1.62 
1.62 
1.62 
1.60 
1.G2 
1.62 

9.882625 

.882887 
.883148 
.883410 
.883672 
.883934 
.884196 
.884457 
.884719 
.884980 

4.37 
4.35 
4.37 
4.37 
4.37 
4.37 
4.35 
4.37 
4.35 
4.37 

10.117375 
.117113 
.116852 
.116590 
.116328 
.116066 
.115804 
.115543 
.115281 
.115020 

39 
88 
37 
36 
35 
34 
33 
32 
31 
30 

31 
32 
33 
34 
35 
36 

9.784612 
.784776 
.784941 
.785105 
.785269 
.785433 

2.73 
2.75 
2.73 
2.73 
2.73 

9.899370 

.899273 
.899176 
.899078 
.898981 
.898884 

1.62 
1.62 
1.63 
1.62 
1.62 

9.885242 
.885504 
.885765 
.886026 
.886288 
.886549 

4.37 
4.35 
4.35 
4.37 

4.35 
407 

10.114758 
.114496 
.114235 
.113974 
.113712 
.113451 

29 
28 
27 
26 
25 
24 

37 

.785597 

2.73 

.898787 

1.62 

.886811 

.0< 

.113189 

23 

38 

.785761 

2.73 

.898689 

1.63 

.887072 

4.35 

.112928 

22 

39 
40 

.785925 
.786089 

2.73 
2.73 
2.72 

.898592 
.898494 

1.62 
1.63 
1.62 

.887333 
.887594 

4.35 
4.35 
4.35 

.112667 
.112406 

21 
20 

41 
42 
43 
44 
45 

9.786252 
.786416 
.786579 
.786742 
.786906 

2.73 
2.72 
2.72 
2.73 

279 

9.898397 
.898299 
.898202 
.898104 
.898006 

1.63 
1.62 
1.63 
1.63 

9.887855 
.888116 
.888378 
.888639 
.888900 

4.35 

4.37 
4.35 
4.35 

4  OK 

10.112145 
.111884 
.111622 
.111361 
.111100 

19 

18 
17 
16 
15 

46 

.787069 

.  <<* 

279 

.897908 

Ian 

.889161 

.OO 
4  OO 

.110839 

14 

47 

.787232 

.  <  ~ 
279 

.897810 

.00 

.889421 

.OO 
4  OK 

.110579 

13 

48 

.787395 

.  i  - 

.897712 

1.63 

.889682 

.OO 

.110318 

12 

49 

.787557 

2.70 

.897614 

1.63 

.889943 

4.35 

.110057 

11 

50 

.787720 

2.72 
2.72 

.897516 

1  .63 
1.63 

.890204 

4.35 
4.35 

.109796 

10 

51 
52 

9.787883 
.788045 

2.70 

9.897418 
.897320 

1.63 

9.890465 
.890725 

4.33 

10.109535 
.109275 

9 

8 

53 
54 

.788208 
.788370 

2.72 

2.70 

.897222 
.897123 

1.63 
1.65 

.890986 
.891247 

4.35 
4.35 

.109014 
.108753 

7 
6 

55 
56 

.788532 
.788694 

2.70 

2.70 

.897025 
.896926 

1.63 
1.65 

.891507 
.891768 

4.33 
4.35 

.108493 
.108232 

5 

4 

57 
58 
59 
GO 

.788856 
.789018 
.789180 
9.789342 

2.70 
2.70 
2.70 
2.70 

.896828 
.896729 
.896631 
9.896532 

1.63 
1.65 
1.63 
1.65 

.892628 
.892289 
.892549 
9.892810 

4.33 
4.35 
4.33 
4.35 

.107972 
.107711 
.107451 
10.107190 

3 
2 
1 
0 

' 

Cosine. 

D  1". 

Sine. 

D.  r. 

Cotang. 

D.  1". 

Tang. 

' 

127° 


141 


COSINES,  TANGENTS,  AND  COTANGENTS. 


141* 


' 

Sine. 

D.  r. 

Cosine. 

D.  1". 

Tang. 

D.  i". 

Cotaiig. 

• 

0 

1 

9.789342 
.789504 

2.70 

2ao 

9.896532 
.896433 

1.65 

9.892810 
.893070 

4.33 

4  OK 

10.107190 
.106930 

60 
59 

2 
3 
4 
5 

.789665 
.789827 
.789988 
.790149 

.  Do 

2.70 

2.68 
2.68 
2  68 

.896-335 
.896236 
.896137 
.896038 

l!65 
1.65 
1.65 

.893331 
.893591 
.893851 
.894111 

.oO 

4.33 
4.33 
4.33 

4  OK 

.106669 
.  106409 
.106149 
.105889 

58 
57 
56 
55 

6 

n 

8 
9 
10 

.790310 
.790471 
.790632 
.790793 
.790954 

2  '.68 
2.68 
2.68 
2.68 
2.68 

.895939 
.895840 
.895741 
.895641 
.895542 

I'M 

1.65 
1.67 
1.65 
1.65 

.894372 
.894632 
.894892 
.895152 
.895412 

.oO 

4.33 
4.33 
4.33 
4.33 
4.33 

.105628 
.105368 
.105108 

.104848 
.104588 

54 
53 

52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.791115 
.791275 
.791436 
.791596 
.791757 
.791917 
.792077 
.792237 
.792397 
.792557 

2.67 
2.68 
2.67 
2.68 
2.67 
2.67 
2.67 
2.67 
2.67 
2.65 

9.895443 
.895343 
.895244 
.895145 
.895045 
.894945 
.894846 
.894746 
.894646 
.894546 

1.67 
1.65 
1.65 
1.67 
1.67 
1.65 
1.67 
1.67 
1.67 
1.6? 

9.895672 
.895932 
.896192 
.896452 
.896712 
.896971 
.897231 
.897491 
.897751 
.898010 

4.33 
4.33 
4.33 
4.33 
4.32 
4.33 
4.33 
4.33 
4.32 
4.33 

10.104328 
.104068 
.103808 
.103548 
.103288 
.103029 
.102769 
.102509 
.102249 
.101990 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 

9.792716 

.792876 

2.67 

9.894446 
.894346 

.67 

9.898270 
.898530 

4.33 

10.101730 
.101470 

39 

38 

23 
24 
25 

26 

27 

.793035 
.793195 
.793354 
.793514 
.793673 

2.65 
2.67 
2.65 
2.67 
2.65 

9  flK 

.894246 
.894146 
.894046 
.893946 
.893846 

.67 
.67 
.67 
.67 

.67 

.898789 
.899049 
.899308 
.899568 
.899827 

4.32 
4.33 
4.32 
4.33 
4.32 

.101211 
.100951 
.100692 
.100432 
.100173 

37 
36 
35 
34 
33 

28 
29 

.793832 
.793991 

xi.OO 

2.65 

f)  CK 

.893745 
.893645 

.67 

ao 

.900087 
.900346 

4.  32 
4  3'* 

.099913 
.099654 

32 
31 

30 

.794150 

/«.  OO 

2.63 

.893544 

.Do 

.67 

.900605 

4^32 

.099395 

30 

31 
32 
33 
34 

9.794308 
.794467 
.794626 
.794784 

2.65 
2.65 
2.63 

9.893444 
.893343 
.893243 
.893142 

.68 

.67 
.68 

9.900864 
.901124 
.901383 
.901642 

4.33 
4.32 
4.32 

10.099136 

.098876 
.098617 
.098358 

29 
28 
27 
26 

35 

.794942 

2.63 

2  OK 

.893041 

.68 

.901901 

4.32 
400 

.098099 

25 

36 

.795101 

.OO 

.892940 

n 

.902160 

.CM 

.097840 

24 

37 

38 

.795259 
.795417 

2.63 
2.63 

.892839 
.892739 

.68 
.67 

.902420 
.902679 

4.33 
4.32 

.097580 
.097321 

23 
22 

39 
40 

.795575 
.795733 

2.63 
2.63 
2.63 

.892638 
.892536 

.68 
.70 
.68 

.902938 
.903197 

4.32 
4.32 
4.32 

.097062 
.096803 

21 
20 

41 

9.795891 

9.8924-35 

9.903456 

4qr» 

10.096544 

19 

42 
43 
44 

.796049 
.796206 
.796364 

2^62 
2.63 

.892334 
.892233 
.892132 

'.68 
.68 

ry(\ 

.903714 
.903973 
.904232 

.oU 

4.32 
4.32 

.096286 
.096027 
.095768 

18 
17 
16 

45 
46 

47 
48 
49 

.796521 
.796679 
.7968-36 
.796993 
.797150 

2^63 
2.62 
2.62 
2.62 

Son 

.892030 
.891929 
.891827 
.891726 
.891624 

.  t(J 

.68 
.70 
.68 
.70 

f»Q 

.904491 
.904750 
.905008 
.905267 
.905526 

4.32 
4.30 
4.32 
4.32 

.095509 
.095250 
.094992 
.0947:33 
.004474 

15 
14 
13 
12 
11 

50 

.797307 

.  DxJ 

2.62 

.891523 

.  Do 

.70 

.905785 

4^30 

.094215 

10 

51 
52 
53 
54 
55 
56 
57 
58 
59, 
60 

9.797464 

.797621 
.797777 
.797934 
.798091 
.798247 
.798403 
.798560 
.798716 
9.79887'2 

2.62 
2.60 
2.62 
2.62 
2.60 
2.60 
2.62 
2.60 
2.60 

9.891421 
.891319 
.891217 
.891115 
.891013 
.890911 
.890809 
.890707 
.890605 
9.890503 

.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 
.70 

9.906043 
.906302 
.906560 
.906819 
.907077 
.907336 
.907594 
.907853 
.908111 
9.908369 

4.32 
4.30 
4.32 
4.30 
4.32 
4.30 
4.32 
4.30 
4.30 

10.093957 
.093698 
.093440 
.093181 
.092923 
.092664 
.092-406 
.092147 
.091889 
10.091631 

9 

8 
7 
6 
5 
4 
3 
2 
1 
0 

1 

Cosine. 

D.  1". 

Sine. 

D.  r. 

Cotang. 

D.  1". 

Tang. 

' 

142 


51° 


TABLE  x. — LOGARITHMIC 


' 

Sine. 

D.  r. 

Cosine. 

D.  r. 

Tang. 

D.  1*. 

Cotang. 

' 

0 

1 

2 
3 
4 
5 
6 

9.798872 
.799028 
.799184 
.799339 
.799495 
.799651 
.799806 
.799962 

2.60 
2.60 
2.58 
2.60 
2.60 
2.58 
2.60 

9.890503 
.890400 
.890298 
.890195 
.890093 
.889990 
.889888 
.889785 

1.72 
1.70 
1.72 
1.70 
1.72 
1.70 
1.72 

9.90a369 
.908628 
.908886 
.909144 
.909402 
.909660 
.909918 
.910177 

4.32 
4.30 
4.30 
4.30 
4.30 
4.30 
4.32 

10.091631 
.091372 
.091114 
.1)90856 
.090598 
.090340 
.090082 
.089823 

60 
59 
58 
57 
56 
55 
54 
53 

8 

.800117 

9  ^ 

.889682 

1.72 

1  79 

.910435 

4.30 

.089565 

52 

9 
10 

.800272 
.800427 

2'.  58 
2.58 

.889579 
.889477 

1  .  (& 

1.70 
1.72 

.910693 
.910951 

4^30 
4.30 

.089307 
.089049 

51 
50 

11 

12 
13 
14 
15 
16 
17 
18 
19 
20 

9.800582 
.800737 
.800892 
.801047 
.801201 
.801356 
.801511 
.801665 
.801819 
.801973 

2.58 
2.58 
2.58 
2.57 
2.58 
2.58 
2.57 
2.57 
2.57 
2.58 

9.889374 
.889271 
.889168 
.889064 
.888961 
.888858 
.888755 
.888651 
.88&54S 
.888444 

1.72 
1.72 
1.73 
1.72 
1.72 
1.72 
1.73 
1.72- 
1.73 
1.72 

9.911209 
.911467 
.911725 
.911982 
.912240 
.912498 
.912756 
.913014 
.913271 
913529 

4.30 
4.30 
4.28 
4.30 
4.30 
4.30 
4.30 
4.28 
4.30 
4.30 

10.088791 
.088533 
.088275 
.088018 
.087760 
.087502 
.087244 
.086986 
.086729 
.086471 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 
22 

9.802128 
.802282 

2.57 

9.888341 
.888237 

1.73 

9.913787 
.914044 

4.28 

10.086213 
.085956 

39 

38 

23 
24 
25 
26 
27 
28 
29 

.802436 
.802589 
.802743 
.802897 
.803050 
.803204 
.803357 

2.57 
2.55 

2.57 
2.57 
2.55 
2.57 
2.55 

.888134 
.888030 
.887926 
.887822 
.887718 
.887614 
.887510 

1  .72 
1.73 
1.73 

1.73 
1.73 
1.73 
1.73 

.914302 
.914560 
.914817 
.915075 
.915332 
.915590 
.915847 

4.30 
4.30 
4.28 
4.30 
4.28 
4.30 
4.28 

.085698 
.085440 
.085183 
.084925 
.084668 
.084410 
.084153 

37' 
36 
35 
34 
33 
32 
31 

30 

.803511 

2^55 

.887406 

l!73 

.916104 

4.28 
4.30 

.083896 

30 

31 

32 

9.803664 

.803817 

2.55 

9.887302 
.887198 

1.73 

9.916362 
.916619 

4.28 

10.0a3638 
.083381 

29 

28 

33 
34 
35 
36 
37 

.803970 
.804183 
.804276 
.804428 
.804581 

2.55 
2.55 
2.55 
2.55 
2.55 

.887093 
.886989 
.886885 
.886780 
.886676 

1.75 
1.73 
1.73 
1.75 
1.73 

.916877 
.917134 
.917391 
.917648 
.917906 

4  .30 
4.28 
4.28 
4.28 
4.30 

.083123 
.082866 
.082609 
.082352 
.082094 

27 
26 
25 
24 
23 

38 
39 
40 

.8047:34 

.804886 
.805039 

2.55 
2.53 
2.55 
2.53 

.886571 
.886466 
.886362 

1  .75 
1.75 
1.73 
1.75 

.918163 
.918420 
.918677 

4!  88 

4.28 
4.28 

.081837 
.081580 
.081323 

22 
21 
20 

41 

9.805191 

2.53 

9.886257 

1  .75 

9.918934 

4  28 

10.081066 

19 

42 
43 

.805343 
.805495 

2.53 

.886152 
.886047 

1.75 

.919191 
.919448 

4  '.28 

49ft 

.080809 
.080552 

18 
17 

44 

.805647 

2.53 

.885942 

1.75 

.919705 

..CO 
49ft 

.080295 

16 

45 

46 
47 

.805799 
.805951 
.806103 

2.53 
2.53 
2.53 

.885837 
.885732 
.885627 

1.75 
1.75 
1.75 

.919962 
.920219 
.920476 

.,*O 

4.28 
4.28 

49ft 

.080038 
.079781 
.079524 

15 
14 
13 

48 
49 

.806254 
.806406 

2.52 
2.53 

.885522 
.885416 

1.75 
1.77 

.920733 
.920990 

.60 

4.28 

49ft 

.079267 
.079010 

12 
11 

50 

.806557 

2.52 
2.53 

.885311 

1  .75 
1.77 

.921247 

..CO 

4.27 

.078753 

10 

51 
52 
53 
54 
55 
56 
57 
58 

9.806709 
.806860 
.807011 
.807163 
.807314 
.807465 
.807615 
.807766 

2.52 
2.52 
2.53 
2.52 
2.52 
2.50 
2.52 

9.885205 
.8a5100 
.884994 
.884889 
.884783 
.884677 
.884572 
.884466 

1.75 
1  .77 
1.75 
1.77 
1.77 
1.75 
1.77 

9.921503 

.921760 
.922017 
.922274 
.922530 
.922787 
.923044 
.923300 

4.28 
4.28 
4.28 
4.27 
4.28 
4.28 
4.27 

10.078497 
.078240 
.077983 
.077726 
.077470 
.077213 
.076956 
.076700 

9 
8 
7 
6 
5 
4 
3 
2 

59 
60 

.807917 
9.808067 

2.52 
2.50 

.884360 
9.884254 

1.77 

.92:3557 
9.923814 

4^28 

.076443 
10.076186- 

1 
0 

'  1  Cosine. 

D.  1". 

Sine. 

D.  r. 

Cotang. 

D.  1'. 

Tang. 

1  . 

129° 


143 


COSINES,  TANGENTS,  AND  COTANGENTS. 


r 

' 

Sine. 

D.  1". 

Cosine. 

D.  1'. 

Tang. 

D.  1". 

Cotang. 

> 

0 
1 
2 
3 

9.808067 
.808218 
.808368 
.808519 

2.52 
2.50 
2.52 
2  50 

9.884254 
.884148 
.884042 
.883936 

1.77 
1.77 
.77 

70 

9.923814 
.924070 
.924327 
.924583 

4.27 
4.28 
4.27 

4OQ 

10.076186 
.075930 
.075673 
.075417 

60 
59 
58 

57 

4 
5 
6 

.808669 
.808819 
.808960 

2^50 
2.50 
2  50 

.883829 
.883723 
.883617 

.  to 

.77 

.77 
70 

.924840 
.925096 
.925352 

.40 

4.27 
4.27 

49ft 

.075160 
.074904 
.074648 

56 
55 
54 

7 
8 
9 
10 

.809119 
.809269 
.809419 
.809569 

2!50 
2.50 
2.50 
2.48 

.883510 
.883404 
.883297 
.883191 

.  10 

.77 
.78 
.77 
.78 

.925609 
.925865 
.926122 
.926378 

,/X> 

4.27 
4.28 
4.27 
4.27 

.074391 
.074135 
.073878 
.073622 

53 
52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

9.809718 
.809868 
.810017 
.810167 
.810316 
.810465 
.810614 
.810763 
.810912 
.811061 

2.50 

2.48 
2.50 
2.48 
2.48 
2.48 
2.48 
2.48 
2.48 
2.48 

9.883084 
.882977 
.882871 
.882764 
.882657 
.882550 
.882443 
.882336 
.882229 
.882121 

.78 
.77 
.78 
.78 
.78 
.78 
.78 
.78 
1.80 
1.78 

9.926634 
.926890 
.927147 
.927403 
.927659 
.927915 
.928171 
.928427 
.928684 
.928940 

4.27 
4.28 
4.27 
4.27 
4.27 
'4.27 
4.27 
4.28 
4.27 
4.27 

10.073366 
.073110 
.072853 
.072597 
.072341 
.072085 
.071829 
.071573 
.071316 
.071060 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 

21 

22 

9.811210 
.811358 

2.47 

9.882014 
.881907 

1.78 

9.929196 
.929452 

4.27 

10.070804 
.070548 

39 

38 

23 
24 

.811507 
.811655 

2.48 
2.47 

24ft 

.881799 
.881692 

1.80 
1.78 

.  .929708 
.929964 

4.27 
4.27 

.070292 
.070036 

37 
36 

25 

26 

27 

.811804 
.811952 
.812100 

.4o 
2.47 
2.47 

.881584 
.881477 
.881369 

1.80 
.78 
.80 

.930220 
.930475 
.930731 

4.27 
4.25 
4.27 

.069780 
.069525 
.069269 

35 
34 
33 

28 

.812248 

2.47 

2  47 

.881261 

.80 
en 

.930987 

4.27 

497 

.069013 

32 

29 
30 

.812396 
.812544 

2^47 
2.47 

.881153 
.881046 

.OU 
.78 
.80 

.931243 
.931499 

,Xt 

4.27 
4.27 

.068757 
.068501 

31 
30 

31 

9.812692 

247 

9.880938 

Q/-V 

9.931755 

10.068245 

29 

32 
33 
34 
35 

.812840 
.812988 
.813135 
.813283 

.4* 

2.47 
2.45 
2.47 

9  4*i 

.880830 
.880722 
.880613 
.880505 

.oU 
.80 
.82 
.80 

OA 

.932010 
.932266 
.932522 
.932778 

4.25 
4.27 
4.27 
4.27 

.067990 
.067734 
.067478 
.067222 

28 
27 
26 
25 

36 

.813430 

<c.4O 

.880397 

.oU 

QA 

.933033 

4.25 

.066967 

24 

37 
38 

.813578 
.813725 

2.47 
2.45 

2AV 

.880289 
.880180 

.oU 

.82 
en 

.933289 
.933545 

4.27 
4.27 

.066711 
.066455 

23 
22 

39 

.813872 

.40 

2  45 

.880072 

.oU 

.933800 

4.25 

49*7 

.066200 

21 

40 

.814019 

2^45 

.879963 

!so 

.934056 

.£( 

4.25 

.065944 

20 

41 

9.814166 

O  AK. 

9.879855 

QO 

9.934311 

10.065689 

19 

42 
43 

.814313 
.814460 

<*.4D 

2.45 

.879746 
.879637 

.06 

.82 

.934567 
.934822 

4.27 
4.25 

.065433 
.065178 

18 
17 

44 

45 

.814607 
-  .814753 

2.45 
2.43 

.879529 
.879420 

.80 

.82 

.935078 
.935333 

4.27 
4.25 

.064922 
.064667 

16 
15 

46 

.814900 

2.45 

2   Aft 

.879311 

.82 

.935589 

4.27 

.064411 

14 

47 

.815046 

.4o 

.879202 

.82 

.935844 

4.25 

.064156 

13 

48 

.815193 

2.45 

24^ 

.879093 

.82 

8.9 

.936100 

4.27 

.063900 

12 

49 

.815339 

.4o 
2  43 

.878984 

.06 

.936355 

4.25 

.063645 

11 

50 

.815485 

2^45 

.87'8875 

.'82 

.936611 

4.27 
4.25 

.063389 

10 

51 

9.815632 

9 

9.878766 

9.936866 

10.063134 

9 

52 

.815778 

00 

.878656 

.83 

.937121 

4.25 

.062879 

8 

53 

.815924 

9  49 

.878547 

.82 

.937377 

4.27 

.062623 

7 

54 

.816069 

c\  An 

.878438 

.82 

.937632 

.25 

.062368 

6 

55 

.816215 

2  .43 

.878328 

.83 

.937887 

'  .25 

.062113 

5 

56 

.816361 

9  4Q 

.878219 

.82 

(DO 

.938142 

.25 

.061858 

4 

57 
58 
59 
60 

.816507 
.816652 
.816798 
9.816943 

2^42 
2.43 

2.42 

.878109 
.877999 
.877890 
9.877780 

.OO 

.83 
.82 
.83 

.938398 
.938653 
.938908 
9.939163 

<  !25 
4.25 
4.25 

.061602 
.061347 
.061092 
10.060837 

3 
2 
1 
0 

' 

Cosine. 

D.  1". 

Sine. 

D.  r. 

Cotang. 

D.  1". 

Tang. 

' 

144 


TABLE   X.— LOGARITHMIC    SINES, 


138° 


' 

Sine. 

D.  1*. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

9.816943 

.817088 

2.42 

o  A.y 

9.877780 

.877070 

1.83 

9.939163 
.939418 

4.25 

10.060&37 
.060582 

CO 
59 

2 
3 
4 
5 
6 
7 
8 

.817233 
.817379 
.817524 
.817668 
.817813 
.817958 
.818103 

2^43 
2.42 
2.40 
2.42 
2.42 
2.42 

.877560 
.877450 
.877340 
.877230 
.877120 
.877010 
.876899 

1  .83 
.83 
.83 
.83 
.83 
.83 
.85 

QO 

.939673 
.939928 
.940183 
.940439 
.940694 
.940949 
.941204 

4.25 
4.25 
4.25 
4.27 
4.25 
4.25 
4.25 

.060327 
.060072 
.059817 
.059561 
.059306 
.059051 
.058796 

58 
57 
56 
55 
54 
53 
52 

9 

.818247 

2A-> 

.876789 

.00 

.941459 

4.25 

.05&541 

51 

10 

.818392 

.'*£ 

2.40 

.876678 

.85 
.83 

.941713 

4.23 
4.25 

.058287 

£0 

11 

9.818536 

240 

9.876568 

QK 

9.941968 

10.058032 

49 

12 

.818681 

,<±i< 

.876457 

.OO 

Oq 

.942223 

4.25 

.057777 

48 

13 
14 
15 
16 
17 
18 

.818825 
.818969 
.819113 
.819257 
.819401 
.819545 

2^40 
2.40 
2.40 
2.40 
2.40 
2  40 

.876347 
.876236 
.876125 
.876014 
.875904 
.875793 

.OO 

.85 
.85 
.85 
.83 
.85 

QK 

.942478 
.942733 
.942988 
.943243 
.943498 
.943752 

4.25 
4.25 
4.25 
4.25 
4.25 
4.23 

4  OK 

.057522 
.057267 
.057012 
.056757 
.056502 
.056248 

47 
46 
45 
44 
43 
42 

19 
20 

.819689 
.819832 

2^38 
2.40 

.875682 
.875571 

.OO 

.85 
.87 

.944007 
.944262 

.XD 

4.25 
4.25 

.055993 
.055738 

41 
40 

21 
22 

9.819976 
.820120 

2.40 

0  QQ 

9.875459 
.875348 

.85 

QK 

9.944517 
.944771 

4.23 

4  OK 

10.055483 
.055229 

89 

38 

23 
24 
25 
26 
27 

.820263 
.820406 
.820550 
.820693 
.820836 

/C.Oo 

2.38 
2.40 
2.38 
2.38 

0  QQ 

.875237 
.875126 
.875014 
.874903 
.874791 

.OO 

.85 
.87 
.85 
.87 

QK 

.945026 
.945281 
.945535 
.945790 
.946045 

.XD 

4.25 
4.23 
4.25 
4.25 

.054974 
.054719 
.054465 
.054210 
.053955 

37 
36 
35 
34 
33 

28 
29 

.820979 
.821122 

<C.OO 

2.38 
2  38 

.874680 
.874568 

.oD 

.87 

Q*V 

.946299 
.946554 

4^25 

400 

.053701 
.053446 

32 
31 

30 

.821265 

2^37 

.874450 

.Ol 

.87 

.946808 

./CO 

4.25 

.053192 

30 

31 

32 

9.821407 
.821550 

2.38 

200 

9.874344 
.874232 

.87 

QK 

9.947063 
.947318 

4.25 

4oq 

10.052937 

.052682 

29 
28 

33 
34 

.821693 
.821833 

.00 

2.37 
207 

.874121 
.874009 

.03 

.87 

QQ 

.947572 
.947827 

.!co 

4.25 

400 

.052428 
.052173 

27 
26 

35 
36 
37 
38 
39 

.821977 
.822120 
.822262 
.822404 
.822546 

•  Of 

2.38 
2.37 
2.37 
2.37 

o  07 

.873896 
.873784 
.873672 
.873560 
.873448 

.OO 

.87 
.87 
.87 
.87 

•   00 

.948081 
.948335 
.948590 
.948844 
.949099 

.Xo 

4.23 
4.25 
4.23 
4.25 

4oq 

.051919 
.051665 
.051410 
.051156 
.050901 

25 
24 
23 
22 
21 

40 

.822688 

-V.  Ol 

2.37 

.873335 

.00 
.87 

.949353 

./CO 

4.25 

.050647 

20 

41 
42 

9.822830 
.822972 

2.37 
207 

9.873223 
.873110 

.88 

9.949608 
.949862 

4.23 

44)0 

10.050392 
.050138 

19 
18 

43 

.823114 

.01 

.872998 

00 

.950116 

.Xa 

.049884 

17 

44 

.823255 

2.35 
2  37 

.872885 

.88 

QQ 

.950371 

4.25 

400 

.049629 

16 

45 

.823:397 

207 

.872772 

.CO 
QQ 

.950625 

./co 
4  go 

.049375 

15 

46 
47 

.823539 

.823680 

.94 

2.35 

.872659 
.872547 

.OO 

1.87 

ICQ 

.950879 
.951133 

./CO 

4.23 

4  OK 

.049121 

.048867 

14 
13 

48 

.823821 

207 

.872434 

.OO 
1  ftft 

.951388 

i.XD 

4  OO 

.048612 

12 

49 
50 

.823963 
.824104 

.01 

2.35 
2.35 

.872321 
.872208 

1  .OO 

1.88 
1.88 

.951642 
.951896 

.29 

4.23 
4.23 

.048358 
.048104 

11 

10 

51 
52 
53 

9.824245 

.824386 
.824527 

2.35 
2.35 

9.872095 
.871981 
.871868 

1.90 
1.88 

9.952150 
.952405 
.952659 

4.25 
4.23 

10.047850 
.047595 
.047341 

9 

8 

7 

54 
55 
56 
57 

.824668 
.824808 
.824949 
.825090 

2.35 
2.33 
2.35 
2.35 

.871755 
.871641 
.871528 
.871414 

1.88 
1.90 
1.88 
1.90 

.952913 
.953167 
.953421 
.953675 

4.23 
4.23 
4.23 
4.23 

.047087 
.046833 
.046579 
.046325 

6 
5 
4 
3 

68 
59 
60 

.825230 
.825371 
9.825511 

2.33 
2.35 
2.33 

.871301 
.871187 
9.871073 

1.88 
1.90 
1.90 

.953929 
.954183 
9.954437 

4.23 
4.23 
4.23 

.046071 
.045817 
10.045563 

2 
1 
0 

Cosine. 

D.  1". 

Sine. 

D.  1".  1 

Cotang. 

D.  1". 

Tang. 

' 

42° 


COSINES,  TANGENTS,  AND  COTANGENTS. 


137" 


• 

Sine. 

D.  1". 

Cosine. 

D.  1".    Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 

9.825511 
.825651 
.825791 
.825931 
.826071 
.826211 
.826351 
.826491 
.826631 
.826770 
.826910 

2.33 
2.33 
2.33 
2.33 
2.33 
2.33 
2.33 
2.33 
2.32 
2.33 
2.32 

9.871073 
.870960 
.870846 
.870732 
.870618 
.870504 
.870390 
.870276 
.870161 
.870047 
.869933 

1.88 
1.90 
1.90 
1.90 
1.90 
1.90 
1.90 
1.92 
1.90 
1.90 
1.92 

9.954437 
.954691 
.954946 
.955200 
.955454 
.955708 
.955961 
.956215 
.956469 
.956723 
.956977 

4.23 
4.25 
4.23 
4.23 
4.23 
4.22 
4.23 
4.23 
4.23 
4.23 
4.23 

10.045563 
.045309 
.045054 
.044800 
.044546 
.044292 
.044039 
.043785 
.043531 
.043277 
.043023 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 
12 
13 
14 
15 
16* 
17 

9.827049 

.827189 

.827328 
.827467 
.827606 
.827745 
.827884 

2.33 
2.32 
2.32 
2.32 
2.32 
2.32 

9.869818 
.869704 
.869589 
.869474 
.869360 
.869245 
.869130 

1.90 
1.92 
1.92 
1.90 
1.92 
1.92 

9.957231 
.957485 
.957739 
.957993 
.958247 
.958500 
.958754 

4.23 
4.23 
4.23 
4.23 
4.22 
4.23 

10.042769 
.042515 
.042261 
.042007 
.041753 
.041500 
.041246 

49 

48 
47 
46 
45 
44 
43 

IS 

.828023 

2.32 

.869015 

1.92 

.959008 

4  .23 

.040992 

42 

19 

20 

.828162 
.828301 

2.32 
2.32 
2.30 

.868900 
.868785 

1.92 
1.92 
1.92 

.959262 
.959516 

4.23 
4.23 

4.22 

.0407*8 
.040484 

41 
40 

21 
22 
23 
24 
25 

9.828439 

.828578 
.828716 
.828855 
.828993 

2.32 
2.30 
2.32 
2.30 

9.868670 
.868555 
.868440 
.868324 
.868209 

1.92 
1.92 
1.93 
1.92 

9.959769 
.960023 
.960277 
.960530 
.960784 

4.23 

4.23 
4.22 
4.23 

10.040231 
.039977 
.039723 
.039470 
.039216 

39 

38 
37 
36 
35 

26 

.829131 

2.30 

.868093 

1  .93 

.961038 

H'  00 

.038962 

34 

27 

28 
29 
30 

.829269 
.829407 
.829545 

.829683 

2.30 
2.30 
2.30 
2.30 
2.30 

.867978 
.867862 
.867747 
.867631 

1  .92 
1.93 
1.92 
1.93 
1.93 

.961292 
.961545 
.961799 
.962052 

4.2o 
4.22 
4.23 
4.22 
4.23 

.038708 
.038455 
.038201 
.037948 

33 
32 
31 
30 

31 
32 

5.829821 
.829959 

2.30 

9.867515 
.867399 

1.93 

9.962306 
.962560 

4.23 

10.037694 
.037440 

29 

28 

33 

.830097 

2.30 

.867283 

1.93 

.962813 

4.22 

4OQ 

.037187 

27 

34 

.830234 

2.28 

.867167 

1.93 

.963067 

.60 

.036933 

26 

35 
36 
37 

38 
39 
40 

.830372 
.830509 
.830646 
.830784 
.830921 
.831058 

2.30 

2.28 
2.28 
2.30 
2.28 
2.28 
2.28 

.867051 
.866935 
.866819 
.866703 
.866586 
.866470 

1^93 
1.93 
1.93 
1.95 
1.93 
1.95 

.963320 
.963574 
.963828 
.964081 
.964335 
.964588 

4^23 
4.23 
4.22 
4.23 
4.22 
4.23 

.036680 
.036426 
.036172 
.035919 
.035665 
.035412 

25 
24 
23 
22 
21 
20 

41 
42 
43 

9.831195 
.831332 
.831469 

2.28 
2.28 

9.866353 
.866237 
.866120 

1.93 
1.95 

9.964842 
.965095 
.965349 

4.22 
4.23 

499 

10.035158 
.034905 
.034651 

19 
18 
17 

44 
45 
46 
47 
48 
49 

.831606 
.831742 
.831879 
.832015 
.832152 
.832288 

2.28 
2,27 
2.28 
2.27 
2.28 
2.27 

.866004 
.865887 
.865770 
.865653 
.865536 
.865419 

l!95 
1.95 
1.95 
1.95 
1.95 

.965602 
.965855 
.966109 
.966362 
.966616 
.966869 

•.JOS 

4.22 
4.23 
4.22 
4.23 
4.22 

.034398 
.034145 
.033891 
.033638 
.033384 
.033131 

16 
15 
14 
13 
12 
11 

50 

.832425 

2.28 
2.27 

.865302 

1.95 
1.95 

.967123 

4!22 

.032877 

10 

51 
52 

KO 

9-832561 
.832697 

oqoQoq 

2.27 

2.27 

9.865185 
.865068 

1.95 
1.97 

9.967376 
.967629 

4.22 
4.23 

10.032624 
.032371 
032117 

9 

8 

7 

OO 

54 
55 
56 
57 
58 
59 
60 

.  OiMOOtJ 

.  832969 
.833105 
.833241 
.833377 
.833512 
.833648 
9.8&3783 

2.27 
2.27 
2.27 
2.27 
2.25 
2.27 
2.25 

!  864833 
.864716 
.864598 
.864481 
.864363 
.864245 
9.864127 

1.95 
1.95 
1.97 
1.95 
1.97  i 
1.97 
1.97 

!  968136 
.968389 
.968643 
.968896 
.969149 
.969403 
9.969656 

4.22 
4.22 
4.23 

4.22 
4.22 
4.23 

4.22 

!  031864 
.031611 
.031357 
.031104 
.030851 
.030597 
10.030344 

6 
5 
4 
3 
2 
1 
0 

' 

Cosine.  1  D.  1". 

Sine. 

D.  1". 

Cotang. 

D.  r. 

Tang. 

' 

ia.fi 


43° 


TABLE   X. — LOGARITHMIC    SIXES, 


' 

Sine. 

D.  1*. 

Cosine. 

D.  1". 

Tang. 

D.  1". 

Cotang. 

' 

0 

1 

2 
3 
4 
5 
6 
7 
8 

9.833783 
.833919 
.834054 
.834189 
.834325 
.834460 
.834595 
.834730 
.834865 

2.27 
2.25 
2.25 
2.27 
2.25 
2.25 
2.25 
2.25 

9.864127 
.864010 
.863892 
.863774 
.863656 
.863538 
.863419 
.863301 
.863183 

1.95 
1.97 
1.97 
1.97 
1.97 
1.98 
1.97 
1.97 

9.969656 
.969909 
.970162 
.970416 
.970669 
.970922 
.971175 
.971429 
.971682 

4.22 
4.22 
4.23 
4.24 
4.22 
4.22 
4.23 
4.22 

10.030344 
.030091 
.029838 
.029584 
.029331 
.029078 
.028825 
.028571 
.028318 

60 
59 
58 
57 
56 
55 
54 
53 
52 

9 

10 

.834999 
.835134 

2.23 
2.25 
2.25 

.863064 
.862946 

1.98 
1.97 
1.98 

.971935 
.972188 

4.22 
4.22 
4.22 

.028065 
.027812 

51 
50 

11 

12 

9.835269 
.835403 

2.23 

9.862827 
.862709 

1.97 

9.972441 
.972695 

4.23 

10.027559 
.027305 

49 

48 

13 
14 
15 

.835538 
.835672 

.835807 

2.25 
2.23 
2.25 
2  23 

.862590 
.862471 
.862353 

1.98 
1.98 
1.97 

.972948 
.973201 
.973454 

4.22 
4.22 
4.22 

499 

.027052 
.026799 
.026546 

47 

46 
45 

16 
17 

18 
19 

.835941 
.836075 
.836209 
.836343 

2  '.23 
2.23 
2.23 

2£)O 

.862234 
.862115 
.861996 
.861877 

l!98 
1.98 
1.98 

.973707 
.973960 
.974213 
.974466 

.JOB 

4.22 
4.22 
4.22 

40O 

.026293 
.026040 
.025787 
.025534 

44 
43 
42 
41 

20 

.836477 

.  4O 

2.23 

.861758   2^00 

.974720 

.iSp 

4.22 

.025280 

40 

21 
22 

9.836611 
.836745 

2.23 

9.861638 
.861519 

1.98 

9.974973 
.975226 

4.22 

10.025027 
.024774 

39 

38 

23 
24 

.836878 
.837012 

2.22 
2.23 

.861400 
.861280 

1  .98 
2.00 

.975479 
.975732 

4.22 
4.22 

.024521 
.024268 

37 
36 

25 

26 

.837146 
.837279 

2.23 
2.22 

.861161 
.861041 

1  .98 
2.00 

.975985 
.976238 

4.22 
4.22 

.024015 
.023762 

35 
34 

27 

28 
29 
30 

.837412 
.837546 
.837679 
.837812 

2.22 
2.23 
2.22 
2.22 
2.22 

.860922 
.860802 
.860682 
.860562 

1.98 
2.00 
2.00 
2.00 
2.00 

.976491 
.976744 
.976997 
.977250 

4.22 
4.22 
4.22 
4.22 
4.22 

.023509 
.023256 
.023003 
.022750 

33 
32 
31 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.837945 

.838078 
.838211 
.838344 
.838477 
.838610 
.838742 
.838875 
.839007 
.839140 

2.22 
2.22 
2.22 
2.22 
2.22 
2.20 
2.22 
2.20 
2.22 
2.20 

9.860442 
.860322 
.860202 
.860082 
.859962 
.859842 
.859721 
.859601 
.859480 
.859360 

2.00 
2.00 
2.00 
2.00 
2.00 
2.02 
2.00 
2.02 
2.00 
2.02 

9.977503 
.977756 
.978009 
.978262 
.978515 
.978768 
.979021 
.979274 
.979527 
.979780 

4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

10.022497 
.022244 
.021991 
.021738 
.021485 
.021232 
.020979 
.020726 
.020473 
.020220 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 

9.839272 
.839404 
.839536 
.839668 
.839800 
.839932 
.840064 
.840196 
.840328 
.840459 

2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.20 
2.18 
2.20 

9.859239 
.859119 
.858998 
.858877 
.858756 
.858635 
.858514 
.858393 
.858272 
.858151 

2.00 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.02 
2.03 

9.98ooas 

.980286 
.980538 
.980791 
.981044 
.981297 
.981550 
.981803 
.982056 
.982309 

4.22 
4.20 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 

10.019967 
.019714 
.019462 
.019209 
.018956 
.018703 
.018450 
.018197 
.017944 
.017691 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

51 
52 

53 
54 
55 
56 
57 
58 
59 
60 

9.840591 
.840722 
.840854 
.840985 
.841116 
.841247 
.841378 
.841509 
.841640 
9.841771 

2.18 
2.20 
2.18 
2.18 
2.18 
2.18 
2.18 
2.18 
2.18 

9.858029 
.857908 
.857786 
.857665 
.857543 
.857422 
.857300 
.857178 
.857056 
9.856934 

2.02 
2.03 
2.02 
2.03 
2.02 
2.03 
2.03 
2.03 
2.03 

9.982562 
.982814 
.983067 
.983320 
.983573 
.983826 
.984079 
.984332 
.984384 
9.984837 

4.20 
4.22 
4.22 
4.22 
4.22 
4.22 
4.22 
4.20 
4.22 

10.017438 
.017186 
.016933 
.016680 
.016427 
.016174 
.015921 
.015668 
.015416 
10.015163 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

~ 

Cosine. 

D.  1". 

Sine. 

D.  1". 

Cotang.  D.  1".    Tang. 

' 

133° 


147 


COSINES,  TANGENTS,  AND  COTANGENTS. 


135° 


f 

Sine. 

D.  1*. 

Cosine. 

D.  1". 

Tang. 

D.  1'. 

Cotang. 

, 

! 

0 

1 

2 

3 
4 
5 
6 

8 
9 
10 

9.841771 
.841902 
.842033 
.842163 
.842294 
.842424 
.842555 
.842685 
.842815 
.842946 
.843076 

2.18 
2.18 
2.17 
2.18 
2.17 
2.18 
2.17 
2.17 
2.18 
2.17 
2.17 

9.856934 
.850812 
.856690 
.856568 
.856446 
.856323 
.856201 
.856078 
.855956 
.855833 
.855711 

2.03 
2.03 
2.03 
2.03 
2.05 
2.03 
2.05 
2.03 
2.05 
2.03 
2.05 

9.984837 
.985090 
.985343 
.985596 
.985848 
.986101 
986354 
.986607 
.986860 
.987112 
.987365 

4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.22 
4.22 
4.20 
4.22 
4.22 

10.015163 
.014910 
.014657 
.014404 
.014152 
.013899 
.013646 
.013393 
.013140 
.012888 
.012635 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

11 
12 
13 
14 
15 
16 
17 
18 

9.843206 
.843336 
.843466 
.843595 
.843725 
.843855 
.843984 
.844114 

2.17 
2.17 
2.15 
2.17 
2.17 
2.15 
2.17 

2  -IK 

9.855588 
.855465 
.855342 
.855219 
.855096 
.854973 
.854850 
.854727 

2.05 
2.05 
2.05 
2.05 
2.05 
2.05 
2.05 

9.987618 
.987871 
.988123 
.988376 
.988629 
.988882 
.989134 
.989387 

4.22 
4.20 
4.22 
4.22 
4.22 
4.20 
4.22 

10.012382 
.012129 
.011877 
.011624 
.011371 
.011118 
.010866 
.010613 

49 
48 
47 
46 
45  1 
44 
43 
42 

19 
20 

.844243 
.844372 

2.15 
2.17 

.854603 

.854480 

2.05 
2.07 

.989640 
.989893 

4.22 

4.20 

.010360 
.010107 

41 
40 

21 
22 
23 
24 

9.844502 
.844631 
.844760 

.844889 

2.15 
2.15 

2.15 

OIK 

9.854356 
.854233 
.854109 
.853986 

2.05 
2.07 
2.05 

9.990145 
.990398 
.990651 
.990903 

4.22 
4.22 

4.20 

10.009855 
.009602 
.009349 
.009097 

39 

38 
37 
36 

25 

.845018 

OIK 

.853862 

.991156 

4.22 

.008844 

35 

26 
27 

28 
29 
30 

.845147 
.845276 
.845405 
.845533 
.845662 

2.15 
2.15 
2.13 
2.15 
2.13 

.853738 
.853614 
.85349(7 
.853366 
.853242 

2.07 
2.07 
2.07 
2.07 
2.07 

.991409 
.991662 
.991914 
.992167 
.992420 

4.22 
4.20 
4.22 
4.22 
4.20 

.008591 
.008338 
.008086 
.007833 
.007580 

34 
33 
32 
31 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

9.845790 
.845919 
.846047 
.846175 
.846304 
.846432 
.846560 
.846688 
.846816 
.846944 

2.15 
2.13 
2.13 
2.15 
2.13 
2.13 
2.13 
2.13 
2.13 
2.12 

9.853118 
.852994 
.852869 
.852745 
.852620 
.852496 
.852371 
.852247 
.852122 
.851997 

2.07 
2.08 
2.07 
2.08 
2.07 
2.08 
2.07 
2.08 
2.08 
2.08 

9.992672 
.992925 
.993178 
.993431 
.993683 
.993936 
.994189 
.994441 
.994694 
.994947 

4.22 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 
4.22 
4.22 
4.20 

10.007328 
.007075 
.006822 
.006569 
.006317 
.006064 
.005811 
.005559 
.005306 
.005053 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

41 
42 
43 
44 

9.847071 
.847199 
.847327 
.847454 

2.13 
2.13 

2.12 

2  -10 

9.851872 
.851747 
.851622 
.851497 

2.08 
2.08 
2.08 

9.995199 
.995452 
.995705 
.995957 

4.22 
4.22 
4.20 

10.004801 
.004548 
.004295 
.004043 

19 
18 
17 
16 

45 
46 

.847582 
.847709 

2.12 

.851372 
.851246 

2.10 

.996210 
.996463 

4.22 

.003790 
.003537 

15 
14 

47 
48 
49 
50 

.847836 
.847964 
.848091 
.848218 

2.13 
2.12 
2.12 
2.12 

.851121 
.850998 
.850870 
.850745 

2.08 
2.08 
2.10 
2.08 
2.10 

.996715 
.996968 
.997221 
.997473 

4.20 
4.22 
4.22 
4.20 
4.22 

.003285 
.003032 
.002779 
.002527 

13 
12 
11 
10 

51 
52 
53 
54 
55 

9.848345 

.848472 
.848599 
.848726 
.848852 

2.12 
2  12 
2.12 
2.10 

9.850619 
.850493 
.850368 
.850242 
.850116 

2.10 
2.08 
2.10 
2.10 

9.997726 
.997979 
.998231 
.998484 
.998737 

4.22 
4.20 
4.22 
4.22 

10.002274 
.002021 
.001769 
.001516 
.001263 

9 
8 
7 
6 
5 

56 
57 
58 
59 
60 

.848979 
.849106 
.849232 
.849359 
9.849485 

2.12 
2.10 
2.12 
2.10 

.849990 
.849864 
.849738 
.849611 
9  84^485 

2  10 
2.10 
2.12 
2.10 

.998989 
.999242 
.999495 
.999747 
0.000000 

4.22 
4.22 

4.20 
4.22 

.001011 
.000758 
.000505 
.000253 
10.000000 

4 
3 

0 

1 

0 

' 

Cosine. 

D.  r. 

Sine. 

D.  1".  |l  Cotang. 

D.  r. 

Tang. 

' 

IJLrrnUlllllllllll 


BIOLOGY 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


